I have never used the adding machine paper but I have done several paper piecing using good ole CHEAP copier paper. I have done it both ways: removed the paper after finishing the block but before sewing to others. Starch in large quantities will be your friend doing it this way! I have waited until the top was pieced together and then removed the paper, then starched the top before sandwiching and quilting.
A great, great, great tip for when you want to remove the paper is to wet the seams with water to moisten the paper and it will come off a lot easier and won't cause you to distort your blocks