tl:dr Better dialogs are possible and in fact have been occasionally taking place, especially when participants try to paraphrase each others’ views. 
Dialog is hard but doable.
The traditional debate format is counterproductive. Luckily, we have some new and better ways to approach disagreements. Daniel Dennett recommends four rules in particular, and the most powerful of them is generously paraphrasing the other participant’s position. In Scout Mindset, Julia Galef refers to this tactic as the “epistemological Turing test”—can you state an opposing position so accurately that you could be mistaken for a proponent of that position? It’s also basically steel-manning, the opposite of straw-manning. See Agreeing How to Disagree.
Starting 2014, Brandon Hendrickson and I developed a number of dialogs as we experimented with how to put these principles into practice. Our best result was a dialog about the value of religion. See Honest Debate: Religion, Good and Bad.
Brandon and I also arranged a dialog on Christianity, which went better than your average debate. Both these debates are by and for atheists, but the format of the dialogs is generally applicable. See Honest Debate: Christianity Good and Bad.
Brandon is a innovative educator, and he boiled down the process to a series of structured exchanges. It really works. Here’s a dialog about white privilege, at the end of which both participants feel as though they have been heard. It starts with a lot of glad-handing in order to establish mutual trust between two strangers. You can fast forward through that part, but if you run a dialog, give this step the time and energy it deserves. See White Privilege Dialog.
Resources
Righteous Mind by Jonathan Haidt is amazing. I’ve run two book discussions on it at my Unitarian Universalist Church, and Haidt’s a fan of Grandmother Fish. His book helps explain why people are so polarized over symbols. See Resources for Studying The Righteous Mind.
Human Swarm by Mark Moffett grounds Haidt’s theories in anthropology and zoology. Roughly, he explains what a “nation” is, for humans, pinion jays, Argentine ants, and a few other species. Can a student of E O Wilson’s do wrong? See my review of Human Swarm.
High Conflict by Amanda Ripley is like the practical take on Haidt’s theory. At the end of the book discussions on Righteous Mind, people would say, “OK, but what do we do?” Ripley has some ideas in that direction, including an account of a liberal religious congregation managing unexpected conflicts among members.
Scout Mindset by Julia Galef zeroes in on the experience of the individual, especially the attitude one needs to cultivate in order to be right more often. If you want to be right, you must be willing to learn that you’re wrong.
The organization Braver Angels organizes structured dialogs across lines of polarization, and they’re worth a look.
The Open Mind platform takes Haidt’s theories and puts them into practice, not as a book but as a program. The introductory module is done individually, and it’s worth undertaking. I’d love to get some experience with this system.
