Note: Below are some of the conclusions I've come to based on experience.
It is difficult to rationally discuss topics where personal belief might be introduced into a conversation. The primary source of contention is always the psychological nature of the belief held by the individual. It is the degree with which the individual identifies with the belief that distinguishes it from being able to be discussed rationally or not. If we consider a belief to be a conclusion an individual has arrived at, then a personal belief could be considered the identification of one's self with that conclusion. Any opposition that might be raised toward the personal belief of an individual is perceived not as an objection to the validity of the belief, but rather as an attack on the identity of the individual who holds it. Losing an argument for the validity of a personally held belief is tantamount to losing one's own identity and can produce cognitive dissonance. In this way personal beliefs are held dogmatically even though they might not be promoted fanatically. The more the individual identifies with the particular belief, the harder it is for them to objectively analyze it. Likewise, the more reliant the individual's identity is upon the belief, the more dogmatically they will hold to it.
The two topics that are most notability ones where people have personal beliefs are politics and religion.
“"Political beliefs are like religious beliefs in the respect that both are part of who you are and important for the social circle to which you belong," said lead author Jonas Kaplan, an assistant research professor of psychology at the Brain and Creativity Institute at the USC Dornsife College of Letters, Arts and Sciences. "To consider an alternative view, you would have to consider an alternative version of yourself."”—Gersema, Emily, Which brain networks respond when someone sticks to a belief?
The fact that people identify themselves with a particular political party or religious sect is the exact reason why they aren't able to objectively discuss either topic to various degrees. The stronger they view themselves as a member of a particular sect, the harder is it for them to see that alternative version of themself. The psychological phenomenon that breeds this type of sectarian thinking is called groupthink.