> I don't know anything about UK law, but what on earth is a "private prosecution"?
The same thing it is in the United States. When you're taken to court by something other than the state.
If Kentucky sues you, it's a public prosecution. If I sue you, it's a private prosecution.
Google could have explained this.
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> In USA you have a criminal prosecution brought by the state.
Not usually, no.
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> Anything else would be a civil lawsuit
A private prosecution is a civil lawsuit if it's a lawsuit.
Not all prosecutions are lawsuits. If I take you to court for breaking a contract, your private prosecution will be a lawsuit. If I take you to court for shooting me, it will be different.