You are not supposed to agree on a prior. That's one of the fundamental insights of the Bayesian inference framework. That different people know different things about a given situation, so they initially disagree, and therefore their priors are different. This should not be surprised. People disagree all the time, and the Bayesian framework just formalizes it.
The different people can then go on and do lots of experiments, collect lots of data and update their priors to posteriors. And the guarantee is that as long as each person's prior was not a mathematically weird function, after enough evidence has been collected all these people will have the same posterior function i.e. they will agree [1].
Exactly. This is why (in my view) scientific research should focus on presenting evidence, not on arguing for certain posteriors or priors. The meta-science process then steers Bayesian beliefs correctly and the evidence-gathering process efficiently. (edit: I see now the top post here on this article also discusses this point.)
The different people can then go on and do lots of experiments, collect lots of data and update their priors to posteriors. And the guarantee is that as long as each person's prior was not a mathematically weird function, after enough evidence has been collected all these people will have the same posterior function i.e. they will agree [1].
[1] The famous Aumann's agreement theorem https://en.wikipedia.org/wiki/Aumann%27s_agreement_theorem is a related result that you might like to read about.