By private I assume you mean a one group spot and not the write your name on the board for your spot amongst 30 other camping pads?
There is some limited private camps on the Ark by the numbers, but they are getting smaller in number every year. If your getting there early (like Thursday night or early Firday) you might be able to snag them but they go quickly from what I have seen.
I would also second the Poudre option, lots of campgrounds of the check in variety and not too on top of each other but fairly regulated and can be crowded like the Ark. Private stuff would be higher up is my guess too, I am having a hard time thinking of anyplace other than the Ark where its river side/free/private spots that can be had.