It is a very interesting dynamic to watch from far.
I teach my kids that we don't wear red. I have 1 red shirt and I wear it on the 4th of July. That is it.
When I meet people in wards and branches where I've lived that went to Utah, I will give them a hard time. I try hard to play the self-righteous role that they expect. I also take the emails from my close friends that are Utah fans when BYU loses.
But because there is a distance, it isn't really that personal. We joke, taunt and harass each other. Then we go home and go back to our families and church callings and get back to work. I, for one, am grateful that I don't live close. It keeps the rivalry fun and competitive, but it never gets ugly.
I have a close friend from HS that has been vilified on here for some things that he said. Did I disagree with him? Absolutely. Did I think he was wrong? Absolutely. Did it change my feelings for him? Not in the least. He is still a good dad, a good friend, and I believe he is a good person. Nothing about the rivalry has changed what I think of him (or anyone else).
Maybe it is because I don't live in Utah.