Can the security staff (equivalent to the U.S. Diplomatic Security Service) of the various Embassies/Missions/Consulates/etc located in New York City carry (concealed) firearms as part of their official duties providing security for their diplomatic staff while not on embassy grounds? Have these rules changed any since the 1960's?
A license is required, see NYC Administrative Code §10-301. This page describes the procedure: the license type is presumably a "carry guard license". You must also get a New York State pistol license. There are various training requirements. Title 38 of the Rules of the City of New York ch. 5 (handgun licenses) says what the police department rules are for licensing, §5-04 describes the Carry Guard License, which basically says you must show the need to be armed and that you have any required licenses. If you can do this, they may issue you a license. You cannot carry a concealed firearm off the job just because you are consular security staff.