So sharing evidence the normal way shouldn't be necessary. Asking someone "what's the evidence for that?" implicitly says, "I don't trust your rationality enough to take your word for it."

I disagree with this, and explained why in Probability Space & Aumann Agreement . To quote the relevant parts:

There are some papers that describe ways to achieve agreement in other ways, such as iterative exchange of posterior probabilities. But in such methods, the agents aren't just moving closer to each other's beliefs. Rather, they go through convoluted chains of deduction to infer what information the other agent must have observed, given his declarations, and then update on that new information. (The process is similar to the one needed to solve the second riddle on this page.) The two agents essentially still have to communicate I(w) and J(w) to each other, except they do so by exchanging posterior probabilities and making logical inferences from them.

Is this realistic for human rationalist wannabes? It seems wildly implausible to me that two humans can communicate all of the information they have that is relevant to the truth of some statement just by repeatedly exchanging degrees of belief about it, except in very simple situations. You need to know the other agent's information partition exactly in order to narrow down which element of the information partition he is in from his probability declaration, and he needs to know that you know so that he can deduce what inference you're making, in order to continue to the next step, and so on. One error in this process and the whole thing falls apart. It seems much easier to just tell each other what information the two of you have directly.

In other words, when I say "what's the evidence for that?", it's not that I don't trust your rationality (although of course I don't trust your rationality either), but I just can't deduce what evidence you must have observed from your probability declaration alone even if you were fully rational.