Incidentally, what Bayesian inferences are you talking about? I thought the usual proposals/methods involved principally reporting log odds, to avoid exactly the issue of people having varying priors and updating on trials to get varying posteriors.

I didn't have any specific examples in mind. But more generally, posteriors are a function of both priors and likelihoods. So even if one avoids using priors entirely by reporting only likelihoods (or some function of the likelihoods, like the log of the likelihood ratio), the resulting implied inferences can change if one's likelihoods change, which can happen by calculating likelihoods with a different model.