That is quite interesting. Until now, I thought that private prosecutions are a feature in most jurisdictions (even if their scope / applicability is restricted). I'd value the right to present a case to a judge, even if the police or prosecution decline to prosecute.
Private prosecution does exist, but only for some crimes. IIRC if the public prosecutor refuses to prosecute libel, I think that is one of the rare few crimes you can prosecute civil-ly.