The so-called “common prior assumption” makes frequent appearances in decision theory, game theory and social welfare theory. The essential problem is that economists (or, indeed, philosophers) do not have a good grasp on what is meant by the word “prior”. When I say, “my prior was x”, what was my first prior on which I begin updating using Bayes or some other rule? Was I born a tabula rasa, identical to all other humans? This seems implausible – newborns do not appear to have identical preferences. What about some point between conception and birth, when we are “empty shells” with the ability to have cognition but without some type of individual preferences? These empty shells can be thought of as similar to Rawls’ original person behind the veil of ignorance.
In this unpublished note deriving from conversations with Aumann, Itzhak Gilboa discusses why, no matter how far back we push, the idea of an identical empty shell is problematic. In particular, we do not wish to allow subjective statements like “I am smarter than person b” because when translated into an objective language for Bayesian purposes, that statement by a is “a is smarter than b” but the statement by b is “b is smarter than a”. But restricting statements of identical empty shells to only objective statements causes a more serious problem. In particular, the statement “I am a” is now excluded, and there are philosophical problems in a person who is not originally self-aware ever learning that they are self-aware. In any case, some axiomatic foundations of priors, similar to what exists for uncertainty and probability, is long overdue (and remains so).
A Fine Theorem 
There is, to my knowledge, no particular reason to think people would have common priors about anything other than special repeatable phenomena such as roulette. It can nonetheless be a useful assumption when we are trying to understand the effects of information and want to rid ourselves of the unpredictable confounding factor of non- common priors.