As of five years ago, less than 10%: http://www.propublica.org/article/by-the-numbers-the-u.s.s-g...

While I agree with your implied conclusion, I don't think your questions completely get us to that answer.

It's possible that despite private prisons being a small minority, with a small minority of the prisoners, they wield a disproportionate influence on the system. That is, in lobbying for whatever policies benefit them, they're willing to accept the result of grossly overburdening the entire system, taking that as collateral damage so that their little corner of it will always be full.

Again, I don't think that's the case. But it could be.

It's not so important how many prisons are for profit, but how much lobbying money is spent furthering their goals.

https://www.washingtonpost.com/posteverything/wp/2015/04/28/...

This is an important point that not that many prisoners are in for-profit prisons, but many of these groups are also lobbying to privatize existing public prisons so they're not an inconsequential influence. They also benefit at the margins a lot. (i.e. if one extra person is incarcerated they're likely to go to a private prisons since the construction of public prisons has slowed a lot).

How many prisons, while not for profit, are largely staffed and have services met by for profit entities?