Not a legal requirement, but CASA will inspect them at audits and compare them to the student's flying record, the trip log, the MR, and the instructor's log book.
One less piece of paperwork on the premises = one less gotcha for a 0.1 error.
It really is easier as a flying school operator to have all the student's paperwork on the premises - it means you can get the admin done after hours etc. Yes, the stude can take it with them if they want.