I think for anyone to be convinced to change their mind, they must have some personal investment in the discussion. I’d venture that for most people, dispassionately finding the truth of a matter doesn’t cut it; at best, they will ultimately be swayed by their unconscious biases despite their best intentions. But if they have a relationship with someone with an opposing view or someone the matter at hand affects, now they have something invested in the outcome of the discussion.
As a concrete example, take gay marriage. There are numerous stories a few years back of people who were against gay marriage softening or drastically changing their position after realizing a family member of theirs was gay and wanted to marry someone of the same gender. I vaguely remember a study also on gay marriage a while back that suggested a door-to-door lobbyist (or whatever the actual word for those kind of people are) had an increased chance of convincing someone to support gay marriage if they first made a personal connection with that person. I believe it was later revealed that there were some methodological issues, but I personally think the general idea is on the right track. Also, I’m sure we all have teachers or other individuals who we have a strong relationship with built on respect for their thought processes.
If you respect, care for, etc. a person, you have a personal investment in that relationship. Whether to maintain that relationship or just because you put more stock in what that person says compared to the average stranger, I’d venture that you’d be more willing to change your mind based on the words of someone you have a relationship with.