Like most people, we were taught to make a case for both sides of a controversy, but I don't think I ever deliberately argue for something I don't believe in 'in real life' - life's too short!
I do think that it's important to really consider where the other 'side' is coming from, though. For one thing, they may be right, and for another, you have to be able to predict their opposition to your point of view if you are going to be able to argue against it. I think that's the stage at which I am most likely to change my mind - if I am wondering what 'they' are going to say, think of something, then can't find a way to counter it, then I really reconsider my own point of view, and sometimes change it altogether. Other times, though, I become further entrenched, as I have more reasons for feeling as I do.
I think that's different from simply 'seeing both sides' but rejecting the other one, and it's very different from sitting on the fence, but maybe I'm saying the same thing in a different way.