When Paper piecing with freezer paper you do NOT stitch through the paper when peicing. The technique I know is you stitch your design on the paper using an unthreaded needle. When you go peice you iron your first fabric unit to the shiney side of the freezer paper, then you add the next peice of fabric to it, right sides together. FOLD the freezer paper back along the preforated line you stitched out previously, then stitch right next to the paper. Open your unit, press, unfold the paper over your newly pressed unit and continue in the same manner with your next piece.
Here is a good step by step tutorial with pictures:
http://www.twiddletails.com/store/in...age=page&id=21
I can't imagine using freezer paper to do traditional paper piecing. In both cases, the paper is removed before sandwiching and quilting.
Actually when I do the freezer paper method I peel off the paper as soon as my unit is done because you can reuse the pattern piece up to 8 times before it quits sticking.
Like BellaBoo, my preferred paper for traditional PP is Vellum. But I have also use newsprint and plain old copy paper.
When cutting out my fabric units I add 1/2" or more to all sides just to be on the safe side and ensure my unit will cover all the parts it is supposed to.