Check again; unless they've changed it this year, they don't ask for specific spots, just dates. They do have closures (e.g., between the bottom of Eight Foot rapid and Ledge Rapid). There are some really sweet spots on river left (Chinle among them), and they tend to be less trammeled than the BLM side.
I usually get a permit for all but the last night - it's worth it for flexibility. Keep in mind, if your last night spot is Slickhorn E, Oljeto or Trimble, you must get a Navajo permit.
Camping on their side without a permit is not cool, and if you get busted it's a nasty little fine. Some parties still do it, unfortunately.