Dont debate, just tell the truth.
What happens when two people both think they are telling the truth, yet contradict one another?
I am really sorry I missed this interesting topic when it was timely. My answer to your question is that they use truth-seeking devices to distinguish false from true, partly true from true, partly false from wholly false etc.
One truth-seeking device is the Toulmin Model of Argument. According to Toulmin, we are obliged to adduce GROUNDS to support our CLAIM, and the GROUNDS and the CLAIM are connected by a WARRANT.
E.g., suppose you and I wish to eat out. I want to have Chinese food and you want Mexican. I argue that we should have Chinese [CLAIM], since we had Mexican last time [GROUNDS] and we had previously agreed to take turns [WARRANT]. You might require me to provide support for my WARRANT by citing its BACKING, which I might do by arguing that taking turns is fair since we are equals. If I feel it necessary, I might provide a QUALIFIER to hint at how sure I am about my CLAIM. I might even mention what kind of REBUTTAL is possible by indicating what circumstances might reduce the strength of my CLAIM. It is usually not necessary for every argument to cite all six parts; many get by on CLAIM, GROUNDS, WARRANT and BACKING.
If both parties work through the Toulmin Model conscientiously, they should be able to figure out where and why they contradict each other. The problem (in my experience) is that most people do not want to bother with grounds, warrants or backing. In fact, they use truth-shunning devices: fallacies in general and ad hominem attacks in particular. Or they simply duck all discussion, which is perhaps the ultimate truth-shunning device.
The Toulmin model is not the only truth-seeking device available. I prefer Christoph Lumer's practical argumentation, but make copious use of Walton's argumentation schemes. The informal logic literature is very exciting.