I've actually been going through this recently with a series of cancer tests with a high false positive rate. When I was discussing it with my specialist he said that he and other doctors don't necessarily think of it in rigorously Bayesian terms, but they're familiar with the sensitivity and specificity of each test and have a pretty strong understanding of what that means based on priors.

Talking to medical students I get the same impression. The refrain I commonly hear is that high sensitivity tests can rule something out, and a high specificity test can rule it in. Obviously that's a reasonably shallow understanding, but when they combine that with their intuition of a prior it seems like they get pretty good results.