Not immediately. The subsidized Stafford loan has a grace period of 6 months and the Perkins loan a grace period of 9 months before the student must begin repaying the loan. When you take a leave of absence you will not have to repay your loan until the grace period is used up. If you use up the grace period, however, when you graduate you will have to begin repaying your loan immediately. It is possible to request an extension to the grace period, but this must be done before the grace period is used up.
If your grace period has run out in the middle of your leave of absence, you will have to start making payments on your student loans.