What kind of connection does Dungeon Deities require to work well?
It doesn't need a good connection at all. I'm pretty sure it only communicates with the server to download a dungeon, to report completion of a dungeon, and perhaps some checks in between in case a player steals one of your powerups while you're playing in another dungeon. But each player plays in their own instance of each dungeon.
The actual dungeon building is done through the dungeon deities website. Tiles, monsters and so on cost gold, and gold is found in other people's dungeons, or earned when a player dies in one of your dungeons.
It was a lot of fun, and it's good to see Error014's making progress!
EDIT: Ninja'd by John. As far as I know, what he says is correct (besides potential for security issues -- I mean, I understand why he said it, but haven't heard any confirmations), but that's just from playing it and talking to Error014.