Bungie has confirmed that you will not be able to trade items or weapons with other players in Destiny.
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Answering a fan question on its site about trading items, Bungie explained that the team decided it wants players to earn each item they have, and transferring to another person is not possible.
"[Trading is possible] only with your enemies. Trade your ammo for their loot. You know the best way to deliver ammo to a Cabal Centurian? You'll figure it out, and it's tons of fun."
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"As for swapping irons in the Tower? Nope. We want you to earn 'em. You should be able to tell a badass story for every sweet jewel in your arsenal. Once you earn them, the various Guardians under your account will be able to trade them, but weapons belong to the players who acquire them through action and bravery."
The developers seem pretty definitive about that one--your items are your own. It's possible that the developers are concerned about exploits and player boosting or real-life currency trading (ask EA Sports about FIFA Ultimate Team), but it could also just be the team's preference.
What is good news is that players will at least be able to trade items among their own characters. This is useful when you get a good item suited to another class that you'd rather not just exchange for Glimmer (the in-game currency), and may give you incentives to level up multiple characters at once.
Destiny launches September 9 on PlayStation 4, PlayStation 3, Xbox One, and Xbox 360.