I want an earth-sized planet with a moon-sized moon to both have complex, possibly intelligent, life. Would it be more realistic for life to develop separately on each body or for life to transfer from one to the other through panspermia? Thanks for your help :)
Scientifically We don't know, but one thing to take into consideration is that if one spread from the other you probably couldn't be sure.
Unless you create wildly different organisms that would have different elements as their basis or having something else instead of DNA they could be similar. It's quite homocentric view, but as far as we know, all life in the universe can share our parts of our DNA (just as we share with fish and bananas), as we don't have other examples that would work.
You should also take in consideration that panspermia from a stellar body is very improbable unless it is heavily targeted by asteroids (very large obstacle for life). Except if it happened when life was starting and at bacterial level at best (most probable for panspermia thanks to some bacteria being able to survive travelling through universe) - their high adaptability gives way for the last point:
Even if life developed from the same seed, in different environments it can evolve into extremely different organisms.
All of this is of course barring the option of intelligent design on the life there, if one was intentionally made to colonise other or both made at the same time
You ask a question that you have answer to. With an earth-size planet and earth moon-size moon you can clearly see that there is no atmosphere on moon [Example: the moon] to have a life developed by itself [Example: the moon] and as we see from the last 60 years it's the life from the earth that try to populate it's moon [Example: the earth].
So: It's more realistic to have life develop on earth-size planet and then transfer to moon.
As others mentioned moon-sized moons are to small to hold an atmosphere. Now you can still protect the moon against solar radiation by enveloping it with the magnetic field of it's parent planet. However the magnetic field of an Earth-like planet isn't nearly strong enough for it.
A Jovian with a larger moon could both evolve life. Very alien life but it would be more likely. Alternatively making twin planets that may or may not share an atmosphere could work. But those would be equal in size.
Now to the second part of your question. Panspermia is I think more likely then simultaneous development. And I wager it comes from the planet to the moon. Completely unlike either way with Earth-like planet and moon. If you want a moon that holds life but isn't just a tidally locked planet go for a frozen moon with a liquid sea underneath.
Again, like a Jupiter and its moons. However such life won't breach the ice and certainly won't look up to the stars and find life on its parent planet.
Life on both bodies is definitely possible, but with potentially a few compromises to make it scientifically kosher.
The biggest thing standing in your way is the size of the moon. The moon is not large enough for an atmosphere, or at least not one similar to the earth's.
The simplest way to make it work would be make the nature of life on the moon radically different from life on earth. For example: you could have an earth-like planet facing a moon where the life is just simple bacteria or have the moon be covered in deep oceans with oceanic sea flora. The difficulty in the ocean life version is that you'd have to find a way to keep it from being an ice moon like Europa (Jupiter) or else the people on the Planet wouldn't be able to easily observe it before travelling there. In any case like these, making the planet and moon have such different types of life would make it up to you to how the life developed; a panspermia situation or a co-evolution situation would both be feasible.
The other way to compromise would be to have it be a binary planet relationship instead of a planet-moon relationship. In this case, with two earth-like planets orbiting each other, it is easily possible to have to planets with similar ecosystems. "Similar ecosystems" being relative of course since when you consider how different life develops on separate continents divided by a salt water ocean over a couple hundred million years, the organisms on planets divided by a space ocean for a few billion years are going to look absolutely nothing alike. The two planets will however be able to both have animal life instead of animal vs bacterial life.
The relative way life would develop on binary planets is largely a matter of statistics. For life on both planets to be complex (animal) life, you're going to want life to develop both independently and simultaneously. If they develop too far apart then life is going to look nothing alike. On our earth, animals have hardly existed for more than half a billion years. For complex life to look up and see complex life, you need them to co-evolve at roughly the same time.
For life to develop independently AND simultaneously, the planets would probably need to have formed simultaneously as well. This would allow life to feasibly develop close to the same time on both planets, give or take a few hundred million years. It would still be unlikely that life would co-evolve like this, but less synchronized their formations, the less likely you are to see life co-evolve. Co-evolution is actually much more likely to occur if life develops through panspermia - not from one plant to the other, but from an external source simultaneously. Say for instance if a comet passes by the pair of planets, breaks up and the remnants spark life on both planets. Life could now reach complexity at very nearly the same time depending on how different the planets are primordially.
For the final part of your question - can co-evolved life developing in this way be simultaneously intelligent? - I'd have to say it's very, very unlikely. Intelligent life evolved and achieved spaceflight in the span of hardly 100 thousand years. It's hard enough for complex life to develop at the same time, let alone for it to be intelligent. I'm not gonna say it's downright impossible, but it's a pretty big stretch. You might need to have a good amount of explanation as to why intelligent life exists at the same time on both planets.
I use this article as source (in French, if someone has one in English, it would be great).
And the answer is no.
Moon isn't large enough to keep an atmosphere. The gravity isn't sufficient to prevent the atmosphere erosion by the Sun radiation. The Moon is too to small to keep a geological activity, and have a magnetic shield like Earth.
It could be possible with a larger planet and a larger satellite (there is example of larger than Earth rocky exo-planet).