A private transmitter was a communications device owned and operated by a private individual or family, rather than a station or starbase. George Samuel Kirk had a private transmitter with the call sign GSK-783. (TOS: "Operation -- Annihilate!")
Doctor Manway, the leader of a Federation science team in the Malurian system, had a private transmitter. In 2267, James T. Kirk questioned whether Lieutenant Nyota Uhura was receiving anything from it, as the Malurians' distress calls had already ceased, but there was nothing. (TOS: "The Changeling")
Later in that same year, Lt. Uhura noted George Samuel Kirk's transmitter's classification as private when Captain James T. Kirk requested that she contact it. (TOS: "Operation -- Annihilate!")