The wiki entry does not look good to me.

Unless you think I'm so irredeemably irrational that my opinions anticorrelate with truth, then the very fact that I believe something is Bayesian evidence that that something is true

This sentence is problematic. Beliefs are probabilistic, and the import of some rationalist's estimate varies according to one's own knowledge. If I am fairly certain that a rationalist has been getting flawed evidence (that is selected to support a proposition) and thinks the evidence is probably fine, that rationalist's weak belief that that proposition is true is, for me, evidence against the proposition.

Consider: if I'm an honest seeker of truth, and you're an honest seeker of truth, and we believe each other to be honest, then we can update on each other's opinions and quickly reach agreement.

Iterative updating is a method rationalists can use when they can't share information (as humans often can't do well), but that is a process the result of which is agreement, but not Aumann agreement.

Aumann agreement is a result of two rationalists sharing all information and ideally updating. It's a thing to know so that one can assess a situation after two reasoners have reached their conclusions based on identical information, because if those conclusions are not identical, then one or both are not perfect rationalists. But one doesn't get much benefit from knowing the theorem, and wouldn't even if people actually could share all their information; if one updates properly on evidence, one doesn't need to know about Aumann agreement to reach proper conclusions because it has nothing to do with the normal process of reasoning about most things, and likewise if one knew the theorem but not how to update, it would be of little help.

As Vladmir_Nesov said:

The crucial point is that it's not a procedure, it's a property, an indicator and not a method.

It's especially unhelpful for humans as we can't share all our information.

As Wei_Dei said:

Having explained all of that, it seems to me that this theorem is less relevant to a practical rationalist than I thought before I really understood it. After looking at the math, it's apparent that "common knowledge" is a much stricter requirement than it sounds. The most obvious way to achieve it is for the two agents to simply tell each other I(w) and J(w), after which they share a new, common information partition. But in that case, agreement itself is obvious and there is no need to learn or understand Aumann's theorem.

So Wei_Dei's use is fine, as in his post he describe's its limited usefulness.

at no point in a conversation can Bayesians have common knowledge that they will disagree.

As I don't understand this at all, perhaps this sentence is fine and I badly misunderstand the concepts here.