That particular sort of disagreement is underlaid by much more fundamental agreement. If you did not share a tremendous amount with the person, such a debate would not be possible. Debate is not a competition in situations where the participants agree upon the fundamental means of determining truth, it is a collaboration. All participants are attempting to reach the truth through contributing facts and reasoning. A conflict would be if one person was of the belief that truth is determined by the intuitive feeling of correctness and another believing it to be determined by rational soundness and correspondence with reality. In that case, neither can contribute anything which helps the other person reach understanding, giving rise to competition.