I’m hazy (at best) on the details, but I have it on good authority that at least one Ohio county allows private attorneys to somehow bypass the normal procedure for arrest warrants. The attorney does something up on his letterhead, offers it to the deputy clerk, who then is required to issue a warrant without any independent inquiry into probable cause. I’ve got a cite to a newspaper article that is supposed to be about this practice, but westlaw isn’t finding it.
I’m not sure if this is what you mean by private prosecution, though…
Several federal laws (notably Clean Water Act and Clean Air Act) allow private citizens to undertake suits against violators of the laws (with various procedures specified). These are not criminal, but civil suits, however.
In the town I work in our court requires it. When there is a complaint of shoplifting we arrest but the store has to sign the complaint. When it comes up for trial the store must have their own prosecutor. This started when the Walmart went in. The township prosecutor was so bogged down with Walmart cases the judge ruled that all shoplifting will be prosecuted by the stores.