It's fine for Bayesian A to choose a prior based on his experience, feelings, and intuitive reasoning. But it's equally reasonable for Bayesian B to assert a completely different prior for her own reasons. Then they will come to different conclusions based on identical evidence.

Except that they're actually using different evidence, because the intuitive feelings that go into a prior are, in fact, evidence. More specifically, the prior is the sum of the outputs of all the heuristics that aren't quite strong or reliable enough to reach conscious awareness, but which nevertheless do, and ought to, influence the probability judgment.