S.L.
I don't know how Private these sites are but I recommend Milo McIver (has a river and kid friendly). Also any campground around Timothy Lake, near Mt Hood and somewhat primative, but more private and very beautiful.
Just saw the comment about Lake Merrill in WA. Yes that is BEAUTIFUL up there...If I recall, it's 1st come-1st served so don't get your hopes up... There are several good campgrounds near Cougar, WA.
Good luck and hope you find something perfect. Happy camping!