Paper should be taunt. I use plastic spring clamps when I use a simple cardboard frame. If I use a paper roll, I clip the paper at the bottom. You want the paper to stay in place as the arrow goes through, not push the paper.
By the way, I spent more time last night on the web looking for the science behind paper tuning, than I ever before. I thought I had it after watching some paradox videos. I thought the key was to get the arrow and bow in tune enough that the fishtailing stayed within the two nodes of oscillation with the feathered end trailing in more or less a straight line, through the paper at 6 feet. I read many explanations of paper tuning and frankly, none made sense and even I knew that some were hogwash theories, and I'm wanting to believe what I've been doing (paper tuning) is worthwhile.
In the end, I was not satisfied. So far, while I believe paper tuning has worked for me for the 30+ years I've used it, I don't know why.
I have a strong acquaintance, archery engineer, that I'm going to ask about this.
If I find that paper tuning is simply a "sugar pill", I'll have to decide if I'm going to keep taking it.