Perhaps a step forward would be for disputants to publicize their probability estimates and update them as the conversation proceeds.

I find myself reluctant to support this idea. I think the main reason is that it seems very hard to translate my degrees of belief into probability numbers. So I'm afraid that I'll update my beliefs correctly in response to other people's arguments, but state the wrong numbers. Is this a skill that we can learn to perform better?

Right now I just try to indicate my degrees of belief using English words, like "I'm sure", "I think it's likely", "perhaps", etc., which has the disadvantage of not being very precise, but the advantage of requiring little mental effort (which I can redirect into for example thinking about whether an argument is correct or not).

ETA: It does seem that there are situations where the extra mental effort required to state probability estimates would be useful, like in the AI-Foom debate, where there is persistent disagreement after an extensive discussion. The disputants can perhaps use probability estimates to track down which individual beliefs (e.g., conditional probabilities) are causing their overall disagreement.