Eiruvin 13b records two years-long disputes between Beit Hillel and Beit Shammai, and surely these are not the only long-running debates. How did that work in practice? Did they really have a single discussion that lasted 2.5 (in one case) or 3 (in another) years, straight through? Or was it something they came back to every now and then, in between other discussions? If the latter, how did they track progress/context (or did they rehash a lot)? Was it normal to have several different "threads" of discourse going at any given time?
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First considering that Gemarra was codified traditionally in 450 more likely in 960 in either case centuries after many of these debates occured, so precise details in how they occurred were not recorded by any person with living memory of them.
However, there is no reason to believe that they occurred any differently than debates amongst Rabbanim(and more likely their students) occur today, just with less speed and no internet coverage. Especially once the Jewish people were divided between the various academies in Eretz Yisrael and Babylonia you would need runners(people who would actually go back and forth and say what was being taught in each) or letters or other correspondence to be sent.
Three debates that I can think of in recent times that took place over a protracted period of time are: Rav Moshe Feinstein and Rav Shlomo Zalman Auerbach over whether chicken bones were muktze(forbidden to be handled on shabbat, unless necessary). That one actually played out between their talmidim for quite some time until the two men wrote to one another and accepted each other's positions for their respective locations(in the US people don't eat chicken bones, and in Israel at the time they did).
Rav Hershcel Shachter vs a whole slew of internet bloggers who are students of big people that have not yet weighed in themselves on the issues regarding agunah/get meusah. This one is still ongoing and getting quite nasty to be honest.