So long as you don't change consoles, yes. All downloads (with limited exceptions) are tied to both your gamertag and the console it was purchased on. So yes, as long as they are playing on the console that originally purchased the items, it should be fine.
(Avatar items are only available to the account itself, and there may be some map packs with the same issue, but that one I don't know for sure.)
And, in some cases, if you're logged in as player 2 or something, things will still work. (It's how my girlfriend used to play PaperBoy when I had to change consoles, but before the DRM tool was available.)
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