I think the OP was asking more about whether you're issued your own personal weapon, as in here, Ensign, this is your pistol; there are many like it, but this one is yours. Answer is no, and the only time you'll carry one is if you're Boat or Boarding Officer or OOD Inport. And even then, you carry it for the duration of your watch/boarding, then hand it off to your relief or return it to the gunner's mates.
If you really want one to call your own, you might be able to score a pretty blue one...
...or mondo red one
070901-N-1810F-234 CARIBBEAN SEA (Sept. 1, 2007) - Cryptologic Technician 2nd Class Michael Whisenant, assigned to a visit, board, search and seizure (VBSS) team aboard guided-missile destroyer USS Mitscher (DDG 57), embarks aboard French frigate FS Ventose (F 733) as part of VBSS training during PANAMAX 2007. Civil and military forces from 19 countries are participating in PANAMAX 2007, a U.S. Southern Command joint and multi-national training exercise co-sponsored with the government of Panama, in the waters off the coasts of Panama and Honduras. U.S. Navy photo by Mass Communication Specialist 2nd Class Todd Frantom (RELEASED)
090404-N-0901C-155 ARABIAN SEA (April 4, 2009) Hull Maintenance Technician 3rd Class Colin Powers and Store Keeper Seaman Daniel Daily sweep a room for suspects during Armed Sentry\Ships Reaction Force Basic (AS\SRFB), a required training for all security personnel. (U.S. Navy photo by Mass Communication Specialist Seaman Apprentice Shonna Cunningham/Released)