Most national forests I know of in the western US forbid camping on active river beds. Folks do drive out on them but I think it's forbidden to camp, and other where you see the driving it's on private property, like on the Eel.

What I'm unsure of is whether you'd get dinged if you parked nearby and just set up tent etc out on the river bed. We've done that frequently boating on those rivers, and it's a more enviornmentally friendly (and sometimes required) place to camp.