Well, you might have different priors, leading to different posterior beliefs from the same data; or you might have different values, leading to different decisions or policy prescriptions from the same descriptive beliefs.
(One might expect that a person raised in a large close-knit extended working-class immigrant family might have different values regarding economics than a person raised in a small individualistic nuclear middle-class ethnic-majority family, for instance.)
Note that I said someone who is more biased in an arena will disagree with me, not that someone who disagreed with me in an arena was exhibiting more bias.
The last point reminded me of speculation from the recent LessWrong article Conspiracy Theories as Agency Fictions:
Before thinking about these points and debating them I strongly recommend you read the full article.