To listen to your opponent is not to hear one statement, judge it as incorrect, and move on to just trying to prove your own point. True listening is not the intent to persuade, but rather the intent to understand. Why would your opponent believe something that is obviously false? They must have reasoning and perspective that you do not have. Each thing ties into something else, and makes a complete system.
Once you understand your opponent's position, it is wise not to just accept it, but to think it over and compare it to Scripture, etc. Then, if trying to also persuade an opponent, one must clearly represent that knowledge of their position, rather than starting with putting forward an assessment of their position that they disagree with.
It seems to me that for the main part, this type of listening is a lost art in America. For the most part, people will listen to a couple points, and if at that point they see an error in your thinking, they will never go on to listen to the whole picture of what you think. They will shoot it down, but will never build a comprehensive mental model of what you actually are trying to say. That style of doing things does not seem to build either clarity or agreement, but rather frustration.
To truly listen is not so much an exercise of patience but of curiosity. Actually be interesting in what they believe (though they are wrong) and why they believe it. In many cases, though they may be wrong about one thing, they may be right about something else. I would go so far as to say that it is likely that there is something you can learn from any person.
"As the Communist atheists allowed no room in their hearts for Jesus, I decided not to leave the slightest place for Satan in mine." -Richard Wurmbrand
Let us all be quick to hear, slow to speak, slow to wrath. Myself included.