Board games by mail are a good analogy for why we need game servers when chance is involved. When you play chess by mail, you can send your moves directly to each other, because there's no chance involved, and nothing is hidden. You don't need to trust that your opponent rolled the dice honestly or shuffled his cards as well as you did. When you do have an element of chance - for instance, drawing tiles in Scrabble - you can't just send your moves to each other directly. You need a neutral third party who manages the bag of tiles, draws them out for you and tells each of you in secret what you have on your rack. This game master takes your move, checks it against what is possible, given the letters in your hand, and passes it on to your opponent if it's legitimate. That's what a game server does online, whether it's for Scrabble, Monopoly, poker or whatever else you are playing. When you're in the same room, you trust each other, at least in part, because you all see what everyone else is doing. You know the dice were rolled properly, so there's no dispute over that. Online, you are separated, so you need a neutral server to handle the chance and secrets for you.
The really interesting question is whether this neutral, trusted third-party role can be spread out among the players. Could you somehow arrange a dice-rolling mechanism where two people each produce part of the result, and they can both trust that the outcome is still based purely on chance?
Mokalus of Borg
PS - I have a feeling it might be possible.
PPS - But I also wouldn't be surprised if it wasn't.