You will not take a bet with Bob. If he does not know the result of the coin, he will not take anything worse than even odds.
You should clearly not offer him even odds. If you offer him anything else, he will accept if and only if he knows you will lose.
Hang on, I just realized there's a much simpler way to analyze the situations I described, which also works for more complicated variants like "Bob gets a 50% chance to learn the outcome, but you get a 10% chance to modify it afterward". Since money isn't created out of nothing, any such situation is a zero-sum game. Both players can easily guarantee themselves a payoff of 0 by refusing all offers. Therefore the value of the game is 0. Nash equilibrium, subgame-perfect equilibrium, no matter. Rational players don't play.
That leads to the second question: which assumptions should we relax to get a nontrivial model of a prediction market, and how do we analyze it?
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.