Who was responsible for determining the MIA's in the Vietnam war?

If a USN Swift Boat is in contact, and during the battle, a crewman is missing; that boat commander forwards the man's name/serial number/circumstances/usally in the After Action Report (AAR), etc., to his higher headquarters; that information continues up the chain of command. The same holds true for any other US unit; if a USAF F-104 Starfighter is shot down (as was LTC Philip Smith's Starfighter in 1965/ref. book: Journey Into Darkness, By LTC Philip Smith), then his unit record's his "Aircraft missing, presumed lost..." pending further disposition. After the facts start rolling in, then that data is processed. Whatever data is accumulated, is forwarded up the chain of command. In Smith's case, he was POW from 1965 until President Nixon got him out of Red China in 1972. Smith's Starfighter had strayed too close to the Red Chinese border during a strike mission over North Vietnam, his jet was engaged by a Red Chinese MIG-19 (J-6 version) and was shot down by an air to air missile.