... surely at least the theorems don't depend on the agents being able to fully reconstruct each other's evidence?
They don't necessarily reconstruct all of each other's evidence, just the parts that are relevant to their common knowledge. For example, two agents have common priors regarding the contents of an urn. Independently, they sample from the urn with replacement. They then exchange updated probabilities for P(Urn has Freq(red)<Freq(black)) and P(Urn has Freq(red)<0.9*Freq(black)). At this point, each can reconstruct the sizes and frequencies of the other agent's evidence samples ("4 reds and 4 blacks"), but they cannot reconstruct the exact sequences ("RRBRBBRB"). And they can update again to perfect agreement regarding the urn contents.
Edit: minor cleanup for clarity.
At least that is my understanding of Aumann's theorem.
That sounds right, but I was thinking of cases like this, where the whole process leads to a different (worse) answer than sharing information would have.
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