There are a couple (at least) of mistakes that novices (say, undergraduates) make about group work.
The first is to divide up the work and then later integrate the pats written by individuals. this actually adds work to the process, rather than simplifying it, since the integration is a significant step.
The second mistake is to assume that everyone takes the same role and does the same thing (on their piece if the above division is also assumed). People work alone on their own part and do the same tasks that "team" members do on theirs.
This is counterproductive.
One of "my" favorite papers was a group work. Two of the team developed the pedagogy and explored it. The rest of us analyzed it and gave them feedback. One of us did the actual writing over several iterations with feedback from everyone else. He happens to be the best writer in the group. Note that everyone took a different role and the role of writing went to the best writer. The author order listed the two that developed the structure as "first".
The second feature of the above is that we all worked together, with frequent (constant) communication. There was no "integration step" and each author played a role best suited to their skills. My role was quite minor overall, though I was able to contribute some insights.
So, my suggestion is to not divide a group project into "chunks" with people taking the same role as everyone else on their chunk, but to divide it into roles that contribute to the whole. Also, to work together, closely together, throughout the process.
If you really want to improve the writing skills of someone else and you have those skills, then I'd suggest that you pair up with them on writing a piece. You do most of the actual writing, and your partner asks questions and contribute things from their primary skill set, whatever it is. Or let them write a bit and comment as it goes along, suggesting alternatives as they go. And, yes, this is really pair programming adapted to writing.
But improving their writing may not be essential for the project if you give appropriate roles to people rather than "chunks".
Hint: You won't improve someones writing (or other) skills by complaining that they do it poorly. Not in the short term anyway, though some kinds of criticism can result in a long term inward look and commitment to change in a person. This has happened to me a couple of times. I recall one in math and another in taichi. But the process it induced in me took a long time to come to fruition. But neither of these was from teammates in any sense, but from teachers. That is their proper role - to induce change and improvement.