The paper is the stabilizer.
Have seen a lot of freezer paper used. In fact I quilted a baby quilt for a friend who used freezer paper and didn't get it all out. I checked with her and she said it would wash soft and be OK ... so I quilted it.
Personally, I'll try a paper towel or go back to stabilizer. Though freezer paper is OK for a class project where I am just learning a technique and not making a work of art.
ali