I didn't say anything about what should happen to your beliefs.

You said that the prior dominating is a bad thing. The prior is your beliefs about a parameter prior to observing data (as I suspect you know!). Maybe I'm not getting what you're saying.

I was assuming these are general purpose statistics, which means you might want to share them with someone. It's bad for those to get tainted by your personal priors. If it's a purely personal calculation then sure that's fine.

That's an interesting observation.

Let's say you have a personal belief that something is going to happen with probability x. Would you actually want to tell others that the probability is y, because that's what the data says, without letting people know that for other reasons that are not reflected in the data, you truly believe it is x?

You can give your informed opinion on the probability in addition to what you've derived from the dataset, but you shouldn't conflate them.

Your informed opinion incorporates this dataset, but you shouldn't imply it's "based on" this dataset.

That's fine and I would agree - you could share the summary statistic used, or the likelihood ratio between the null and some alternative models.

But you shouldn't share a frequentist parameter estimate or confidence interval if you have prior information that would influence it non-negligibly, at least not without sharing that prior information also.

Should you have any beliefs in the absence of data? And if you have some prior data but no additional data now why carve out past as separate thing and call it prior? Why not just call everything you have - data?

That's precisely how Bayesian inference works! But rather than having to repeat all analysis of prior data sets, we summarize that analysis in the form of a posterior, which becomes the prior for the next analysis.