Let's say I'm playing coop with someone, and we want to play a custom map that only I have. Is it possible, perhaps through the console, for me to send them the file and then load it without either of us leaving the game?

Nope. Point them to where you downloaded the file. Each client needs the (same!) map file.

I guess that eliminates the possibility of playing custom maps with random partners.

But given that Valve said custom maps would be playable on consoles as well, do you think they're going to implement some sort of in game gallery/browser? In that case, it might be easy to share maps.

I'm sure they have something in store for this problem, because it's definitely an issue. I know that in normal multiplayer games like TF2, the server sends the map to the client. I actually tested trying to do this method in Portal 2 Coop but sadly it didn't work. I don't think having a global repository of maps would be the best solution, I think just having the host send the map to the client would work the best and require the least work on Valve's end.

All in due time.

Yeah. One of the things that made the original Half Life and every Valve game since so ripe for modding was the ability to distribute custom maps on the fly. I see no reason why this game will be any different, they just have to tackle features one at a time.