I have always been fascinated by these kinds of situations. I try to put myself in their shoes and see how or if I could react better.
This is clearly a pissing contest about who can speak over the other for the longest period of time. It accomplishes nothing, makes blood boil, and makes it incredibly hard for anyone to hear what you said even well after the spat.
So ... what's the distorted incentive that makes this happen? I would propose that it is machista bravado. If either backs down they appear weak and they don't want to appear weak in front of a third-party (the audience). How can you get out of that situation without looking weak then?
I would probably start off the same way, talking over the other, but after about three seconds and after the other person says the obligatory "can I finish?" I would respond "you have ten seconds, then let me say my piece" and shut up. Then I would countdown from 3 down (probably with hand signals and I would make it take longer than 3 seconds). After that I would just ask "when can I talk?" every three seconds until they shut up.
I'm not saying this is the most mature way to handle it, but it's better than that 30 seconds of nothing happening.
How would you respond in that situation?