Suppose that your good friend, the police commissioner, tells you in strictest confidence that the crime kingpin of your city is Wulky Wilkinsen. As a rationalist, are you licensed to believe this statement? Put it this way: if you go ahead and insult Wulky, I’d call you foolhardy. Since it is prudent to act as if Wulky has a substantially higher-than-default probability of being a crime boss, the police commissioner’s statement must have been strong Bayesian evidence.
This thread started with one of the examples you came up with.
It’s not privileging the hypothesis to Aumann-agree with someone. See Scientific Evidence, Legal Evidence, Rational Evidence:
In general, people know what experiences they’ve had, and can exchange information about them through talking with each other. Strong evidence is common.