Ah! You're not the first to be confused here. So - the multiplayer is entirely Words With Friends-style asynchronous -- you get your opponent's entire turn once they hit submit, but not their individual moves as they come in. (You can quit the game in the middle of a match and come back.) This probably needs a popup explaining it the first time. There's an experimental synchronous mode that I've disabled in this build because it's buggy, and because I'm not convinced it would be healthy for the game (since it would require a bigger player pool), but we're still making up our minds about that...
Sounds good. It would still be nice if it had a status for the other player: {offline, active, idle} but I'm sure it's not a top priority.