Some things not to overlook:
For most topics of debate, there are valid, reasonable arguments for both sides. Just because you find one side more convincing does not mean the other side has an untenable position!
This point is overlooked by most people -- they seem to think that anyone who presents an argument opposing their beliefs must automatically be dumb, deceived, deluded... anything but reasonable.
Coming to a reasoned position on a topic requires one to analyze all sides of the topic in good faith -- in particular, this requires knowledge and understanding of the arguments favoring each individual side.
This is another thing that people tend to overlook. A person might be predisposed to adopting position A. When they go about studying the arguments for position B, they notice flaws or rebuttals to those arguments. Unfortunately, they are inclined to stop there and declare position A convincing, without ever taking the effort to follow-up and make sure that the flaws or rebuttals they noticed can stand up against scrutiny.
Finally, the model of two sides presenting a case to an arbiter really is used in the "real world". Two examples:
1. In principle, this is how a trial in the U.S. is supposed to work -- two sides presenting the best cases for the two opposing positions, and the judge or jury evaluating the given evidence to arrive at a conclusion. (Also, note the prosecution is supposed to give the defense any evidence it finds that could help its case. The reverse doesn't happen, but that's in accordance with te principle that it's supposed to be harder to prove someone guilty than to defend someone as innocent)
2. This process was used by the Roman Catholic Church in the process of sainting people -- the job description of the Devil's advocate was to present an argument why the candidate shouldn't be sainted. (and his opponent, God's advocate, was to present the argument in favor)