Peer Disagreement: What Should I Believe?

If a peer of equal intellectual capability and insightfulness disagrees with me, should I abandon my view, hold fast to my view, or at least hold my view with greater tentativeness and lesser certitude?

What if my intellectual peer and I are serving on a jury and we review the same evidence? My peer thinks the person on trial is guilty, but I think he's not guilty. Even though my intuition tells me one thing, should I lessen a hold on my belief since my intellectual peer on the jury is persuaded otherwise?

Philosophers of equal stature disagree on a lot of things--God's existence, free will vs. determinism, the soul's existence, whether abstract objects like numbers exist, what knowledge or truth is, whether life has meaning or not, what political theory to follow, whether objective beauty exists, and so on. So given all of these disputed matters by equally intelligent persons, what should I do? Should I just give up and be agnostic or neutral about everything? How could I even make important life decisions when others of equal intellectual stature would disagree with my decision on a number of things?

Peer disagreement is much-discussed in philosophical circles, and it has proven to be a very engaging topic. How can we get a foothold on this topic? I trust we can make some headway in our discussion--even if we don't see eye to eye!