Work Reform

You're right that only a minority of inmates are held in private facilities:

In fact, less than 9% of all incarcerated people are held in private prisons; the vast majority are in publicly-owned prisons and jails. Some states have more people in private prisons than others, of course, and the industry has lobbied to maintain high levels of incarceration, but private prisons are essentially a parasite on the massive publicly-owned system — not the root of it. Nevertheless, a range of private industries and even some public agencies continue to profit from mass incarceration. Many city and county jails rent space to other agencies, including state prison systems, the U.S. Marshals Service, and ICE. Private companies are frequently granted contracts to operate prison food and health services (often so bad they result in major lawsuits), and prison and jail telecom and commissary functions have spawned multi-billion dollar private industries. By privatizing services like phone calls, medical care, and commissary, prisons and jails are offloading the costs of incarceration onto incarcerated people and their families, trimming their budgets at an unconscionable social cost.

Source

However, this does nothing to make the fact that private individuals profit from the incarceration of people any better. This practice could hardly be more unethical or counterproductive, as it creates a financial incentive to incarcerate as many people as possible—and it is obvious that this runs counter to the social purpose of prisons.