In practice, it doesn't seem to make much difference:

> Propensity score matching and regression analyses of the National Inmate Survey are used to examine individuals’ experiences and perceptions of environmental quality in public versus private prisons across key domains of prison life, with an emphasis on needs, behaviors, victimization, and attitudes. No differences are identified for women. Males in private prisons are more likely than men in public prisons to perceive there to be inadequate staffing and are less likely to report maintaining ties with people outside of prison. Men in private prisons are also less likely to report crowding to be an issue.

https://www.researchgate.net/publication/361439680_Private_V....

What we really need is a prison voucher program. The prisoners get to go to whatever prison they want to, and then the prison they go to gets paid. Incentives aligned!

Does it? The alternative is non-profit incarceration/detention, in which the incentive is to not put people in prison, leading to more crime, lower trust and more victims.