Is it possible? Yes, why not. Will there always be Sun in the way? No or at least not for Earth or any planet in our system. Does it matter? No and here is why.
I simplified a few things, I'm using Kepler Laws (assuming elliptical orbits and none interference from other bodies) and round some things and completely disregard relativity. Relativity would not play that big role anyway, becasue the time lag to the other planet would be about 17 minutes for light.
I ran the calculations for Earth and assuming we would be at aphelion (farthest from sun) when the other planet were at perihelion (closest to sun), the sun would not be in our way to see each other for some portion of the year (here is where the calculation gets complicated because of calculation of areas of parts of ellipses, but there could be some iterative ways to calculate it. It is not the main point of the question so I won't).
Somewhere in mid-spring/mid-fall (best day would be the fall solistice or two days after or during the spring solistice or two days before, because two days are about the time difference between exactly half orbit of our ellipse on aphelion and perihelion halves) when you would look up, you would see the planet. Except you would have to look up at day. And the planet would be about one degree from the sun. In astronomy one degree is often approximated by outstretching your hand and extending your pinky. That's about the distance you would be able to see the planet from the sun.
Now that all was based on Earth, which has eccentricity of about 0.0167 and because of our distance to sun the distance between the two foci of our orbit are about 5 million kilometers apart. But let's take a look at a different planet of our system, Venus.
With eccentricity of only 0.0067 and considerably smaller orbit, its foci are only about 1,4 million kilometers apart, which is about the diameter of Sun (it's important to point out that Sun's center is in one of the foci, so the Sun almost touches the center of Venus' orbit). That would mean that when the Sun would not be directly in the way, it would be so close that it would not mostly matter and the path would be there for only brief time, about 12 hours if we make some approximations.
That would mean there would be a window for 12 hours twice a year in which you would have the chance to see the other planet, assuming you have some means to completely shut out the luminosity of the Sun. Based on our civilization, we would probably find our neighbor by accident when investigating other places in our Solar system, which gets us to another rather interesting astronomical phenomenon and that is...
Lagrangian points. Those are basically five points where in a system of two large bodies (e.g. Sun and Earth) a small enough object stays stationary or orbits around the Lagrangian point (see Halo orbits). Yes you guessed it, the third one is exactly oposite to the smaller object behind the bigger (for Sun and Earth it would be exactly your scenario).
The drawback is the great instability of the third point and the size of the objects, we know about some, but for Earth they are usually asteroids of size about 1 km or smaller. I have not heard about a planet forming in a Lagrangian point of another one, but then again, I've never researched that, but it would surelly mean that one of them would be a giant and the other a dwarf. And it is important to note here, that the two bodies would actaully not share the exact same orbit, so it would not hold for this question either.
All in all, it would be possible and for it to happen there are three main parameters: the planets' eccentricity, the stars diameter and the planets' orbital distance to the star.