Ideas matter. There are correct ideas, and wrong ideas. This is the agreement implicit in any sincere debate. Without this agreement, the content of any debate is irrelevant. (Of course, there are also social interactions where content doesn't matter -- but they are not debates, precisely.) Of course, the debate itself is only considered useful if there is a mismatch of knowledge between participants. If they think the same, they have nothing to learn but the fact that they think the same. And if they think differently, it means that at least one has something to teach to the other, that the other doesn't know, or currently disagrees with.
Of course, every sincere person believes her ideas are correct and that opposite ideas are therefore wrong. (Note however that different ideas are not necessarily opposite.) In open debates without an important prize to the winner (such as money, power, sex), such as happen in public forums, debaters are usually sincere, and will indeed promote ideas they believe are correct. They may believe the other party is a hypocrite, but that is seldom correct. Double standards and other forms of irrational thinking are usually a better explanation of why a party takes a stance that is in contradiction with what the other party thinks she may logically be thinking. Of course, such irrational attitude may actually be followed by either party, or both.
Now, if someone sincerely believes something about a topic one cares to debate, and tries to convince the other party of it, it does not mean that one isn't ready to possible changing one's ideas. But that's not the goal of a discussion between knowledgeable persons. Persons who talk about things they care for have thought about those things; they have developed strong mental structures of the considered domains. And that's precisely what makes the debate interesting. You don't want to debate with a person who hold random opinions for no specific reason -- if that's a habit, the person is utterly mindless, and if that isn't then it means the person doesn't care much about the question. Therefore, in an interesting discussion, the two parties will each have a more or less elaborate but in any cases strong mental structure of the domain being debated. Unless there's some fragile part in the structure that had been oversighted and can be stumbled upon during the discussion, there is little chance of changing anyone's mind in the debate. It is thus stupid to take changing one's or the other party's mind as the normal goal of a debate.
The goal of a debate is to understand the other party better, and get better understood by the other party. If agreement follows, it's all the better. If disagreement endures, oh so well. But in either case, the discussion will have been fruitful if mutual understanding has progressed -- or at least if unilateral understanding has progressed -- or maybe if only reflective understanding has progressed -- still, if understanding has progressed. This understanding may progress as each party unfolds the mental structure through which one envisions the discussed domain, and as the other party can internalize a similar structure and see how the bits fit or don't fit his own.
Often, one just fails to correctly understand some pieces of the other party's structure, and it is necessary to backtrack to more elementary parts. Sometimes, a lot of backtracking is necessary before a common basis is found upon which to start discussion. When people have different cultural backgrounds, when they have different temperaments, it may require quite a bit of effort, on at least one party, so as to find a common language. Sometimes, the effort is not worth it, and it is better to cease the discussion. But if the parties are ready to each take the time and effort required to further the debate, then it may be fruitful and yield richer understanding.