Some time late last year, I somehow accidentally did a guy a favor by letting him get to Sanctuary by joining my game. Apparently he really appreciated it, because he invited me to his game, powerleveled me to 30, and gave me a Conference Call and some mod (the one you get for beating... some kind of epic-type dude. I don't play often, don't know the name). I remember leaving them in the safe, but when I came back few weeks ago, all my cool high level stuff (that I still can't use anyway, my best character is a level 34 assassin) was gone. So did they get rid of duped items?
Any modded, non-legit items that you keep in your character's personal inventory will get deleted after you quit. However, if you put a modded item in your vault storage, the game doesn't check there and it will remain, you just have to remember to drop it off before you exit the game.
Your question regarding duped items doesn't pertain to this. A duped item that could legitimately be obtained (it is actually in the game) will remain even if it was obtained through illicit means (duping). It only checks for items created with stats that aren't in the game's loot tables.