There was a lady on one of my groups who did all of her paper piecing by hand and it was a BUGGER to remove the paper from her swap blocks. She used regular weight copy paper and a very tiny, double threaded stitch. She used hand quilting cotton thread so it didn't break but if she would have used regular machine piecing thread, I think it would have.
If you're going to paper-piece by hand, use thin paper or iron some muslin or light interfacing onto freezer paper, print the pattern off onto the muslin/interfacing (remove freezer paper after printing) and sew onto that. No paper to remove and a nice solid foundation to work with.