I haven't done a 180 in a single conversation, but I (nice, white, middle class liberal) made a lot of conservative friends at university and they made me rethink a lot of things. Before that, I had never really had my beliefs on things like gay marriage or abortion challenged - I didn't actually believe that any right-minded person could possibly object if they actually thought about it. Now I know better. I'm still a pretty bleeding heart liberal but I have reconsidered several things over the years, like tax incentives for businesses and fox hunting.
Re: armed police
The first time I saw a policeman with a gun was when I was 19 and on holiday, I think in Italy. We walked into a square and there were two of them casually there with guns strapped to their belts in plain view and I FREAKED. I thought the only reason there could be armed police in the area is if they were actually hunting down some dangerous criminal and we were about to be caught in the middle of a gunfight. It was explained to me that all Italian police carry guns as a matter of course, but I still remember my first reaction of total fear - what if they shot me by accident? I cannot imagine living somewhere where your average bobby on the beat has a lethal weapon. It would be horrific. As someone has said above, police shootings of civilians or civilians killings of police officers are rare in the UK and reported very seriously. There is a plaque near my parents' house to a policeman who was killed in the eighties and people still talk about it in the local area.