A while back, David Brin wrote up a proposal for "disputation arenas".

An interesting part of the competition/argument process:

Phase Four: The Paraphrasing Challenge

Here is a unique, but crucial phase. After both manifestos are declared logically usable, a distinct period -- say a month -- will be given to each side so they may paraphrase the other side's position.

This step aims to ensure that each party has actually read and understood where the other one stands, so they aren't simply shouting past each other at chimeric caricatures. Paraphrasing is hard to do when you've spent years demonizing the opposition, calling them venal or stupid and dismissing their concerns. Success at paraphrasing will be seen as a way of winning credibility. It means "I do understand my opponents... so my disagreement with them is well-informed."

Each side may lace their formal paraphrasing with asterisks and footnotes asserting that the statements they are describing are idiotic. That's fine. But if they fail to depict their opponents' point of view in a manner that the jury finds at least generally accurate, they will be disqualified.