The most common example, and also the worst, of privatization of government functions occurs with the prison industry. More than any other, "private prisons" are held up as a reason why government functions should not be privatized.
Perhaps the reason is that with other functions, sometimes the job gets done. Construction projects to eventually get completed and many contracting functions are also examples of experts being brought in to give advice on a task. With prisons there is little in the way of advanced technical expertise, and the job is ongoing.
Still, the "private prison" is not private. Every single customer of a "private prison" is a government agency. Every single client of a “private prison” is sent there by a government agency.
A private prison would have private customers, those that subject people to incarceration outside of the government system. A private prison would have private clients, people incarcerated outside of the government system. Since those don’t exist, every single example of a so-called private prison is not actually private but is instead a contracted prison.
This also means that if there was a private prison, the owner and all of the employees would be criminals. They would all be engaged in the crime of kidnapping, as well as other assorted crimes necessary to make the private prison operate.
There is no real justification for people to make this kind of error. Private and contracted are two different arrangements. Just because the company that conducts the enterprise makes a profit does not mean that it is a private enterprise.