In my experience, the important part of writing things jointly is to give everyone the time and space to write at their own pace. This means that one of the most awkward things to do is to send around a single word document to which everyone then adds a version number. That's because this mode means that only one person can write at any given time -- independent of whether that person has the time to do that right now, or whether someone else actually just had a good idea worth putting into the document. It also inevitably leads to situations where someone edits an older version and then some poor soul has to figure out what the changes were and port them forward to the current version -- further blocking everyone else's progress.
So whatever mode you adopt, make sure you have a way to ensure that everyone can write whenever they want. MS Word documents just don't allow for that, but that's not a problem: There are many other ways you can use, and in some of these cases the final version can be converted either into MS Word, or into another format that can then be submitted to whomever needs to get the document in the end. Examples that I have used in the past are:
For things that don't require extensive formulas, just use a Google Document. Everyone can add and edit whenever they want to add and edit, including concurrently while someone else is working on another section. The final version can then be exported in MS Word, OpenOffice document, or PDF format.
If you need formulas but don't know latex, just use Overleaf.com. Again, everyone can add and edit whenever they want and concurrently if necessary. The end result can be exported as PDF.
If you know latex, you can also just work on the plain latex files and share them via github. Github has mechanisms to resolve conflicts if several people started writing at the same time, starting from the same version, but commit them at different times.
Of these options, github is probably the most awkward mechanism because your changes are not recorded immediately, but only when you upload them (possibly causing conflicts with changes others have made). The advantages are that you can work offline, and that you retain a perfect version history if you ever need to find out who wrote which line. The latter seems to rarely be a concern when writing papers, proposals, presentations. The former is a concern, but is becoming less of one as everyone tends to be online all the time these days.