This answer will be divided into several parts.
In this story, I would want two planets that are both large enough to support life, but don't have to be the same or close in size.
Do you want your planets to be the right size to support carbon-based liquid water using life in general, or the more specifica case of humans beings (and other oxygen breathers) in particular?
Part one: The size range for habitable planets in general.
It is beleived that there are a number of subsurface oceans beneath the ice covered surfaces of some moons and dwarf planets in the outer solar system. And there could possibly be life in those subsurface oceans, getting energy from volcanic vents on the ocean floor. Enceladus, a moon of Saturn, with an average radius of 252.1 kilometers and thus an average diameter of 504.2 kilometers, and a mass of only 0.000015 that of Earth, is the smallest of those worlds which might possibly have lifeforms in subsurface oceans.
So Enceladus has only 0.039 of the average radius. 6371.0 kilometers, and the average diameter, 12,742 kilometers, of Earth. Earth has about 25.25.27 times the diameter of Enceladus.
Of course it is possible for worlds larger than Earth to be habitable and have life.
A superhabitable planet is a type of exoplanet or exomoon that may be better suited than Earth for the emergence and evolution of life.
Heller and Armstrong proposed that a series of basic characteristics are required to classify an exoplanet or exomoon as superhabitable;78 10 for size, it is required to be about 2 Earth masses, and 1.3 Earth radii will provide an optimal size for plate tectonics. In addition, it would have a greater gravitational attraction that would increase retention of gases during the planet's formation.10 It is therefore likely that they have a denser atmosphere that will offer greater concentration of oxygen and greenhouse gases, which in turn raise the average temperature to optimum levels for plant life to about 25 °C (77 °F). A denser atmosphere may also influence the surface relief, making it more regular and decreasing the size of the ocean basins, which would improve diversity of marine life in shallow waters.
A superhabitable planet might or might not be habitable for humans. It wouldn't be habitable for humans or beings with similar environmental requirements if the surface gravity was too high or if it is a water world without any dry land, But it would be habitable for lifeforms with similar biochemestry to lifeforms of Earth, and more so that Earth is habitable for Earth life.
So the diameters of worlds habitable for carbon based liquid water using life forms could range from 0.039 the diameter of Earth to at least 1.3 times the diameter of Earth, a range of at least 33.33 times, and possibly much greater.
On the other hand, the Wikipedia article on planetary habitability says:
The radius of a potentially habitable exoplanet would range between 0.5 and 2.5 Earth radii.
Part two: the size range of planets habitable for humans in particular.
But what if you want the planets to be habitable for human beings or beings with similar environmenal requirements?
In that case the source to check is Habitable Planets for Man, Stephen H. Dole, 1964.
In chapter Four The Astronomical Parameters, Dole discusses the properties of a habitable planet first.
On page 53 Dole says that for a planet to have less than 1.5 g, the surface gravity of Earth, and thus be habitable for humans, it should have a mass of 2.35 Earth masses, a radius 1.25 that of Earth, and an escape velocity of 15.3 kilometers per second.
On page 54 Dole calculates that for a planet to retain oxygen in its atmosphere for geological eras of time and be habitable for humans, it should have an escape velocity of at least 6.25 kilometers per second. And that would correspond to a planet with a mass of 0.195 Earth mass, a radius of 0.63 Earth radius, and a surface gravity of 0.49 g.
Thus the possible radii of habitable planets could range over 1.98 times.
But Dole does not believe that a planet with a mass that low could form a breathable oxygen rich atmosphere in the first place. On the following pages Dole decides the minimum possible mass for a planet to form a breathable atmoshere would be between 0.25 and 0.57 Earth mass, and chooses 0.4 Earth mass as the minimum mass for a planet habitable for humans and similar life forms. Such a planet would have a radius of 0.78 Earth radii and a surface gravity of 0.68 g.
So if Dole is correct about that, the greatest possible range in diamenters of habitable lanets would be about 1.6 times.
Another discussion of habitabiity, "Exomoon Habitability Constrained by Illumination and Tidal Heating", Rene Heller and Roy Barnes, Astrobiology, Volume 13, number 1, 2013, discusses the mass range of habitable worlds in Section 2. Habitability of exomoons.
A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (MsT0.1M4; Tachinami et al., 2011); to sustain a substantial, long-lived atmosphere (MsT0.12M4; Williams et al., 1997; Kaltenegger, 2000); and to drive tectonic activity (MsT0.23M4; Williams et al., 1997), which is necessary to maintain plate tectonics and to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Gurnett et al., 1996; Kivelson et al., 1996), suggesting that satellite masses > 0.25M4 will be adequate for considerations of exomoon habitability. This lower limit, however, is not a fixed number. Further sources of energy—such as radiogenic and tidal heating, and the effect of a moon’s composition and structure—can alter the limit in either direction. An upper mass limit is given by the fact that increasing mass leads to high pressures in the planet’s interior, which will increase the mantle viscosity and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed around 2M4 (Gaidos et al., 2010; Noack and Breuer, 2011; Stamenkovic´ et al., 2011).
So they consider the mass range for habitable worlds to range from about 0.25 to 2.0 Earth mass. Since 0.25 Earth mass is more than Dole's 0.195 Earth mass minimum, and 2.0 Earth mass is less than Dole's 2.35 Earth mass maximum, the diameter range for those worlds should be less than 1.98 times.
All things considered, a writer concerened with scientific plausibility should avoid making one planet habitable for humans more than twice the diameter of another planet habitable for humans or similar lifeforms.
Also see the answers to this question:
And I will return to this answer later and discuss how large twin habitable worlds would look in their skies.
And in the meantime you should look at the answers to this question:
Part three: How large could the components of a double planet look in their skies?
In my answer to the question:
I discussed the largest possible angular diameters of various objects as seen from the skies of habitable planets. But I didn't discuss how large the components of a double planet might look from each other's skies, which may have been an oversight on my part.
The Roche limit of an astronomical object is how close another object can approach it without being tidally disrupted. It depends on the relative masses and densities of the two object.
If the Moon was fluid, it would be disrupted at a distance of 18,331 kilometers from Earth. Since the moon is rigid, it would be disrupted at a distance of 9,492 kilometers.
An Earth like habitable planet could orbit closer to another Earth like habitable planet without being tidally disrupted, since both would be much more dense than the Moon.
I think that the Roche limit is measured from the central point of the larger bodies. Since the radius of Earth is 6,371.0 kilometers, if the central points of the two planets were 9,492 kilometers apart the surfaces of the two planets would overlap by about3,250 kilometers. Adding 3,250 kilometers to 9,492 kilometers increases the distances between centers to 12,742 kilometers. Adding another 2,000 kilometers so seach atmopshere can be 1,000 kilometers high without interacting, and the centers of the two planets would be at least 14,742 kilometers apart.
[Added nov. 14, 2021. According to my calculations if one planet is viewed from a place on the other planet where itis on the horizon, it will have an angular diameter of 46.745 degrees, while if it is fiewed from the closest spot on the other planet it will have an angular diameter of 74.584 degrees.]
Thus the center of one planet would be about 8,371 kilometers above the head of someone on the point of the other planet which was directly facing the first planet. The angular resolution of the typical human eye is about 1 arc minute, or about 2.4350 kilometers, or 2,435 meters, at a distance of 8,371 kilometers.
But most of the curface of the planet would be much closer than 8,371 kilometers from the person viewing it from the point directly facing it. At the closest distance, the planetary surface "above" would be about 2,000 kilometers away, and at that distance one arc minute would about 0.5789 kilometers, or about 579 meters.
By comparison, when the Moon is closest to Earth, at about 362,600 kilometers, one minute of arc is about 105.476 kilometers. So people should normally be able to see features with constrating colors that more than 105 kilometers wide on the Moon with the naked eye.
So that seems to be approximately the best possible view of a sister Earth like planet from its partner in a twin planet. If the two planets are separated by greater distances the view will not be as good.
Of course it takes a lot of time for a newly formed Earth like planet to become habitable. I believe that it took about four billion years for Earth to develop a breathable atmosphere with a high oxygen content. Some alien planets might take much longer to become habitable for humans, and some might take much less time, but it is reasonable to suppose that it takes several billion years for an Earth like planet to become habitable.
Of coures if a highly advanced civilization discovers a young Earth like planet, they might decide to terraform it to make it habitable billions of years before it otherwise would become habitable. And I guess they could do it with a double planet as well.
And this is important because tidal interactions between a planet, its moon, and its star, will cause the orbits to change very gradually, which will add up to big changes over billions of years. So by the time a planet becomes habitable, the orbits of the planet and its moons, if any, will have changed drastically from what they were originally. And the same goes for the components of a double planet orbiting their star - such a situation could be considered to be one extreme of aplanet and its moon.
It has long been calculated that the Moon must be receding from the Earth, and experiments with bouncing lasers off the reflectors left on the Moon by Apollo astronauts confirm it.
Thus the distance between Earth and Moon is increasing, and the Earth's rotation is slowing in reaction. Measurements from laser reflectors left during the Apollo missions (lunar ranging experiments) have found that the Moon's distance increases by 38 mm (1.5 in) per year (roughly the rate at which human fingernails grow). Atomic clocks also show that Earth's day lengthens by about 15 microseconds every year, slowly increasing the rate at which UTC is adjusted by leap seconds. Left to run its course, this tidal drag would continue until the rotation of Earth and the orbital period of the Moon matched, creating mutual tidal locking between the two. As a result, the Moon would be suspended in the sky over one meridian, as is already currently the case with Pluto and its moon Charon. However, the Sun will become a red giant engulfing the Earth-Moon system long before this occurrence. If it were to happen, the rotation of the earth would continue to slow down because of the tides caused by the sun. With the day longer than the month, the moon would move slowly from west to east in the sky. The tides caused by the moon would then cause the opposite effect from before, and the moon would get closer to the earth. Eventually it would come within the Roche limit and be broken up into a ring.
But the Sun is predicted to swell up into a red giant star, and engulf Mercury and Venus, and probably the Earth and the Moon, in "only" about five billion years, which shoud be many, many billions of years before the Moon stops receding and begins to get closer to the Earth.
At a rate of 38 milimeters per year, the moon should recede 38,000,000 milimeters per million years, which is 38 kilometers per million years, or 38,000 kilometers per billion years. So in four billion years the Moon should have receded 152,000 kilometers, which is about 0.395 of the semi-major axis of the Moon's orbit.
But when the Moon was much closer to Earth, it would have receeded faster.
When a planet captures another object and makes it a moon, that Moon can orbit the planet either prograde, in the same direction as the planet orbits, or retrograde, in the opposite direction from the rotation of the planet.
Tidal interactions between a planet and a retrograde moon will cause the retrograde moon to slowly approach the planet and eventually break up or crash into the planet. MOst retrograde moons in our solar system orbit far from their planets, and the tidal effects on their orbits are very slight. But triton, the large moon of Neptune, orbits in a retrograde orbit close enough that it is predicted that Triton will be destroyed in "only" about 3.6 billion years.
The fate of prograde moons depends on whether their initial orbits are above or below the distance of a planetary synchronous obit.
Prograde moons which originate outside or above the synchronous orbit will recede from the planet until they either pass outside the planet's HIll sphere and escape from the planet, or slow down the planet's rotation until the planet is tidally locked to the moon. Then tidal interactions with the star in he system will cause those moons to gradually approach their planets until they get too close and are destroyed.
Prograde moons which originally orbit inside or below the synchronous orbit will gradually approach the planet until they get too close and are destroyed. There are over a dozen such moons in our solar system, some of which might have existed in a sub synchronous orbit for over 4 billion years, slowly approaching certain doom.
The moon with the shortest life expectency is probably Phobos, the inner moon of Mars:
Tidal deceleration is gradually decreasing the orbital radius of Phobos by two meters every 100 years., and with decreasing orbital radius the likelihood of breakup due to tidal forces increases, estimated in approximately 30–50 million years, with one study's estimate being about 43 million years.
And the tidal interactions between two components wiht identical mass of a double palnet should be merely one example of a planet-moon relationship. One one extreme mas ratio, a planet has no moon and thus no tidal interactins with it. On the other extreme of the mass ratio spectrum, the planet and the moon have exaactly the same mass, thus making it not only a double planet but a twin planet.
So if a forming planet spilts into two planets which then orbit each other, or if two wandering planets capture each other and become a twin planet, their tidal interactions should be similar to those of a planet and its moon.
If the two planets form together, they will orbit each other in a prograde direction. If they form inside their mutual synchronous orbit, they will spiral closer to each other until they eventually destroy each other. If they form outside their mutual synchronous orbit, they will gradually recede from each other until they escape from each other or both are tidally locked. If both become tidally locked, they will gradually start to approach each other and eventually destroy each other.
If the two planets form separately and later capture each other, they might orbit each other either in a prograde or a retrograde orbit. If they orbit in a prograde orbit, the possibilities in the previous paragraph will apply to them.
If the captured planets orbit each other in a retrograde orbit, they will gradually approach each other until they get too close and destroy each other.
So no matter what distance the two planets are at when they start to orbit each other, that distance will change. They will either get closer and eventually destroy each other, or they will get farther and father apart until they either escape from each other or begin to move back toward each other.
And it should take a very long time, billions of years, for a planet to become habitable with an oxygen rich atmosphere. During those billins of years, if the two planets in a double planet are approaching each other, they should get too close and destroy each other. And if the two planets are getting farther away for billions of years while they become habitable, they should reach a distance of hundred of thousands or millions of kilometers apart before they become habitable and/or intelligent life evolves on them.
So if the two planets in a double planet are spectaculary close togther with a great view view of eachther, they should either destroy each or move much farther apart, resulting in a more ordinary view, long before they become habitable for humans or intelligent life evolves on them.
So a story could have humans in space suits exploring a very young double planet with the two planets very close to each other. Or an advanced civiization might terraform a very young double planet.
Or possibly the two planets, whether formed together or captured later, initially orbit each other at exactly their mutual synchronous orbital distance. I suppose that if the two planets orbit at exactly their mutual synchronous orbital distance tidal interactions will not change the distances between them. And if they orbit very slightly closer or farther than their mutual synchronous orbital distance their orbital distance will change very slowly over billions of years.
I don't know how to calculate how far apart two Earth like planets orbiting at their mutual synchronous orbital distance, but it should be close enough that the smallest features on one planet seen with the naked eye from the other planet should be only a few times larger than in my best case calculation above.
Part Four: the brightness of a double planet.
Since the two planets in a close double planet wold certainly be tidally locked to each other, Planet A would always be visible in the near hemisphere of planet B, in day or night. Planet a would never be visible from the far hemisphere of planet B, and so would never lighten the night of the far side of planet B.
And the same goes for planet B as seen from planet A. Planet B would never be seen from half of planet A, and so would never brighten the night of that side of planet A.
The semi-major axis of the Moon's orbit is 384.399 kilometers, although the Moon has an elliptical orbit getting both closer to and farther from Earth. Because of the changng distance between the Earth and the Moon, the angular diameter of the Moon as seen from eArth varies between 29.3 to 34.1 arc minutes.
The equatorial radius of the Moon is 1,737.4 kilomreters, or about 0.2727 that of Earth's 6,371 kilomeeters. Thus the Earth has 3.667 times the diameter of the Moon. The Earth as seen from the Moon has 3.667 times the angular diameter of the Moon as seen from the Earth, and thus has 13.446 times the angular area of the Moon as seen from Earth.
Since the Moon has an agular diameter about the same as a dime held at arm's length, an angular diameter 3.667 tiems as great would still be appear rather small. But observers on the moon should see large features of land and ocean with the naked eye, and notice large storms.
So the Earth as seen from the Moon would have 13.446 times the brightness of the Moon as seen from Earth. Because of the greater albedo or reflectiveness of the Earth, about 3.36 times as great, the Earth is about 45 times as bright as the Moon. So since the full Moon has an apparent magnitude of - 13 from the Earth, a full Earth has an apparent magnitude 43 times as great, or about - 17.
Imagine that two Earth size planets are about one fifth as far apart as the Earth and the Moon, with a semi major axis of about 76,879.8 killometers. Each one will have 18.335 times the angular diameter and 336.17 times the area of the moon. If they are about 3.36 times as reflective as the Moon, they will appear about 1,129.53 times as bright as the Moon. So an Earth sized planet when full would have an apparent magnitude of about -20.5.
Suppose that they are about one tenth the separation of the Earth and the Moon, with a semi-major axis of about 38,439.9 kilometers. Each one will have 336.67 times the angular diameter and 1,344.7 times the area of the moon. If they are about 3.36 times as reflective as the Moon, they will appear about 4,518 times as bright as the Moon. So an Earth sized planet when full would have an apparent magnitude of about -22.
Suppose that they are about one fifteenth the separation of the Earth and the Moon, with a semi-major axis of about 25,626.6 kilometers. Each one will have 55 times the angular diameter and 3,025 times the area of the moon. If they are about 3.36 times as reflective as the Moon, they will appear about 10,164 times as bright as the Moon. So an Earth sized planet when full would have an apparent magnitude of about -23.
Suppose that they are about one twentieth the separation of the Earth and the Moon, with a semi-major axis of about 19,219.95 kilometers. Each one will have 73.34 times the angular diameter and 5,378.75 times the area of the moon. If they are about 3.36 times as reflective as the Moon, they will appear about 18,072 times as bright as the Moon. So an Earth sized planet when full would have an apparent magnitude of about -23.5. That would be abut 4.5 percent of the brightness of the Sun.
Of course that is comparing the planets in their full phases to the full Moon. When the planets are not full they will be much less bright.
I note that the closer the planets are the brighter they will be when full, and the more likely they will be to be eclipsed when they are full. The closer the planets are, the wider the shadow of each will be in space, and the longer the other planet will be in that shadow.
And if the two planets orbit each other in almost exactly the same plane as they orbit their star, they should eclipse each other every night When a planet is eclipsed and receives very littl light from its star, it can reflect very little light back at the planet which is eclipsing it.
So if the two planets orbit very close to each other, each should be eclipsed by the other during each period of daylight on the side facing the other one, and each should eclipse the other one during night time on the side facing the other, and thus should greatly reduce the the amount of light the other one reflects back at it to light up the night.
And of course the sides of the planets facing away from their partners will have regular alterations of uneclipsed daylight and dark, starlit nights without any planet shine lighting them up.
Thus it may be pssible to find a distance between the two planets which allows much detail on their surfaces to be seen with the naked eye, while allowing the nights to be dark enough.