Great idea, there are so many games that would actually be played with even a basic co-op support like this. It's okay it's just an overlay with no shared-game state, still really fun!
I developed a modded emulator for Pokemon Red/Blue that allowed multiplayer (as in, multiple people walking around in the world, all interacting with the npcs and being able to battle each other) but unfortunately when I asked around there seemed to be no interest. I still have it on github, maybe some day I'll get back into it.
Oh but there is shared game state! Most of SRAM data is synced for ALTTP and there's even a PvP feature so you can slash at each other with swords and use some of the items to damage each other. Of course you can enable/disable whatever bits of state to sync or not.
https://github.com/Salgat/PokeSynch