Writing process in a group
There is no universal process for writing papers.

Some faculty members don't touch a manuscript until the student has written a draft. Others are intensively involved in the planning process before the paper writing process starts. Still others largely write most of the paper themselves. It really is a function of the individual advisor's tendencies. 

However, I think the optimal use of everyone's time and expertise—and to improve the student's development, since the student will ultimately have to write her own papers someday, is to follow a strategy of "taking the training wheels off" over time. For a first paper, the advisor should be very intensely involved throughout the process of organizing and writing. As the student gains experience and learns to work more independently, less and less direct involvement and input from the advisor occurs—although some involvement at the planning and writing stages is best. For the last papers the student writes, she should be the primary driver of the process.