Exchanging items during movement phase
Basically, if you pass through the same location as someone else, you can give up or receive items from them. As long as it's not combat, there are no timing requirements. This is described on page 23 of the rulebook. The rules for trading items are quite generous:


  Investigators have the ability to trade Common Items, Unique Items,
  Spells, and Money, as long as they are in the same street area,
  location, or Other World location. They can do this in any phase, and
  do not need to end their turn in the same area in order to do so.
  However they can not trade items during combat.
  
  Allies, clues, and trophies may not be traded.


Note especially that for gameplay reasons you can trade Spells, but not Allies or clues. From a thematic perspective this has always seemed a little counter-intuitive to me (You teach someone else the spell, and then... forget it? "But of course my friend can't go off with you!" "Yep, I have a clue, but I simply can't tell you about it.").