Can you exchange items with another player, or can you only give items to another player on your turn?
The updated rulebook states (on page 7, under that same heading "Using Items and Omen Cards") that an item can be given, dropped, or picked up. This is much clearer: the other player can't give you an item if it isn't their turn. (During the Haunt, you may attack an opponent to steal an item in lieu of doing damage, as long as you would have inflicted 2 or more damage. But that's not what you're talking about.)

On a side note, the FAQ clarifies that each item is only going to have something happen to/with it once per player turn. So you can't both use and give a weapon to repeatedly attack with it when you have a pile of Heroes wailing on a monster.


  For each item, during a turn an explorer or monster that can carry items may only perform one of the following actions: use the item, give the item to another explorer, drop the item, steal the item, or pick up the item. Using the item means making any attack or roll or other action in which the item is involved in any way.