Anne and Bob are two Bayesians who initially share a non-degenerate prior about a binary state of the world. Anne observes some signal (i.e., an experiment in Blackwell's terminology) about the state of the world. (Formally, a signal/experiment is a joint distribution on the state of the world and the data that Anne observes.) Observing this signal induces some distribution of Anne's posterior beliefs. Denote that distribution on $[0,1]$ by $A$.
Bob also observes a signal (not necessarily the same as Anne's, not necessarily conditionally independent from Anne's, etc.) about the true state of the world. Let $B$ denote the distribution of Bob's posterior beliefs.
What can we say about the relationship between random variables $A$ and $B$? For example, are $A$ and $B$ necessarily (strictly) positively correlated? Are there any other restrictions on their joint distribution?
I specified a binary state of the world so supports of $A$ and $B$ are in $[0,1]$ and it is easy to talk about their correlation, but any results that apply to an arbitrary state space would also be of great interest.