I heard private rafting trips are possible. What are the terms of that option? Meaning do you have to have a certain size group to secure? Extra fees?

Yes you can have a private group, but you still need a guide for the trip.

Noc has self guided rafting and ducky boats. You are responsible to remember which side of the rapids to be in to avoid the potential flipping or rock crashing! If you're okay with that go for it! I've done 3 already this year. 40 years of canoeing. Ymmv. Have fun!

Private trips are indeed traditional. There is a public putin and a public takeout. However, your question informs a basic lack of experience. Talk to NOC or Endless Rivers, another outfitter, or a whitewater club that provides instruction. You'll be better served.

Definitely have guided trips throughout the day. Idk the minimum size group. I'd call for pricing