Marginal Revolution linked to A Fine Theorem, which has summaries of papers in decision theory and other relevant econ, including the classic "agreeing to disagree" results. A paper linked there claims that the probability settled on by Aumann-agreers isn't necessarily the same one as the one they'd reach if they shared their information, which is something I'd been wondering about. In retrospect this seems obvious: if Mars and Venus only both appear in the sky when the apocalypse is near, and one agent sees Mars and the other sees Venus, then they conclude the apocalypse is near if they exchange info, but if the probabilities for Mars and Venus are symmetrical, then no matter how long they exchange probabilities they'll both conclude the other one probably saw the same planet they did. The same thing should happen in practice when two agents figure out different halves of a chain of reasoning. Do I have that right?

ETA: it seems, then, that if you're actually presented with a situation where you can communicate only by repeatedly sharing probabilities, you're better off just conveying all your info by using probabilities of 0 and 1 as Morse code or whatever.

ETA: the paper works out an example in section 4.