I routinely am a co-author on papers with 10-16 authors, as is the norm in my field (biology/molecular biology; I just counted, and in the past 12 months, about a dozen papers, I averaged 11.1 authors). The process I use is by no means ideal, but it is very much a standard in my fields.
With more than three or four authors, it's not expected that most will be heavily involved in the writing process. A "middle" author -- that is, one other than the last author (senior) and first author (who did the bulk of the work) -- typically gets a near-final draft and is asked for input on that. They'll often find a block in the Methods and the Results section commenting something like "JESSICA - please fill in details of mouse examinations" or something like that, and we expect a detailed couple paragraphs there, but the implication is that aside from their specialty, we're looking for typos and minor changes and not much more.
(If a middle author has serious concerns, of course we pay attention. But if the draft is accurate and reasonably well written, it's not expected that a middle author will offer major revisions.)
That means that you're really doing the bulk of the manuscript prep with two, or perhaps three, main writers, which makes life a lot easier; which is important, because what just about everyone does is pass Word documents back and forth and use "Track Changes". Sometimes we use shared folders (Dropbox equivalents); sometimes we use email. In either case we typically keep a list of drafts; one set of revisions come back, we save as "v3" and then "Accept revisions" and call that version "v4", and so on.
We never work simultaneously. One person makes all their comments, tells the next "Your turn", and waits until all those changes are made.
Eventually, usually after three or four passes, this turns into a draft that can go out to middle authors. Here, because we don't expect major changes, we typically do get simultaneous changes, and just incorporate them all together.
We often send to the couple of authors who we know are likely to make more changes first (Jessica is really good at identifying logical flow problems; James is British and has language eccentricities), incorporate changes from them, and after that send to the people who are likely to just catch typos.
Again, this is all using Word Track Changes, which I know has many problems, but it's a de facto lowest common denominator; everyone has it, everyone knows how to use it, it's easier than trying to teach one (or 11) people how to use Git or whatever. More computation-heavy fields probably use different approaches. When I'm a middle author on a paper, this is what I expect, and it's what I invariably get. It sounds like it should be a mess, but since it's what everyone is used to it generally goes pretty smoothly. It's been years since the logistics of working with co-authors has been an issue for me.