They don't start with the same priors or preferences, but unless they have a good reason to trust their own more than the other person's, they'd use both.

For example, imagine you were in some treasure hunt. Everyone has a different, imperfect map. This could result in you all looking in different places, but if you stop and think about it (and it's not a competition) you'll just share maps. The fact that one map happened to be in your hands doesn't make it better.