I can tell you one arrangement that I don't like so much, used by someone I collaborated with once. The idea was that there would be a LaTex file containing everything we had, all results, maybe some speculations, etc. At any given time one of the coauthors would be designated the "editor" and would have control of the file and be responsible for adding new material to it. Depending on who was most active at the moment, or who was more familiar with the subject of interest at the moment, editorship could change.
It sounded like a good way to do things, with the clear advantage that one always had an accessible summary of what had been done up to that point. But in the end I felt that this method tended to bias the final paper toward a particular style --- an "everything but the kitchen sink" paper. Whereas my preference, usually, is for clarity and elegance, to the extent one can achieve this. And that might mean leaving minor results or dead ends out.
The coauthored papers I'm happiest weren't developed in any special way. Just emails would be sent back and forth, and sometimes you would have to dig through old emails to find something done earlier. When we felt the project was complete one person would volunteer to write it up, and then the other or others would suggest changes. I think this is a personal thing and there's no one right answer.