It was government prisons nearly all of those people were put into during the prime years of the massive drug war, not private prisons. Commenters always and without exception intentionally ignore that fact, for obvious reasons (implodes the propaganda). I'm entirely against private prisons, however it's comical the lengths people will go to, to pretend it wasn't government prisons those millions of people were put into. The prison boom was a government prison boom. The prison industrial complex was a government effort almost exclusively. The drug war was a government effort. The cops (government) benefitted, the police unions benefitted, the government contractors benefitted, the politicians benefitted (jobs and construction in their districts, along with more tax money and government spending).
Are you making the claim that private prisons reduce total inmates? If not, this claim, truth or not, is irrelevant.
Also, to your point, private prisons only account for 8.41% of the prison population in the U.S. Link: https://en.wikipedia.org/wiki/Private_prison#United_States