We use this system as well. I have spliced eyes in the ends of the sheet and use a soft shackle to connect the ends together. I've tried to run them through a large plastic ring attached to a bunji at the back of the trampoline to keep them clear but my crew prefers to have the spin sheets loose on the trampoline. He manages them well and they are rarely a problem so I don't interfere.

I think the length of the spin sheet is key here, though. If they're too long, they'll wash off the side. They need to be JUST long enough that the crew can get his foot in the back footstrap while sheeting the kite at a deep angle...no longer.