Wouldn't the easiest way to see if that would work would be to test it on a ROM you have? I think that it wouldn't work as well as you except. I don't think that in Generation III there was 4 way battles between trainers (feel free to correct me if I'm wrong). But if it is in the game I think that if you did some serious hacking (like ASM) you could trick the game into thinking some computer trainers are human trainers and you could have the four way battle.
So yes, I think you both can walk around, but I don't think you two could battle together or interact a lot.