Exchanging Cards in Pandemic
From the rules:


  Share Knowledge Sometimes it’s hard
  for one player to get the cards
  necessary to discover a cure. The Share
  Knowledge action (while difficult to
  perform) can be useful in these cases.
  Transfer a card from one player to
  another. Every card transferred costs
  1 action. Both your pawn and your
  fellow player’s pawn must be in the
  same city, and you may only transfer
  the card of the city that you are in
  together. (For example, if you are
  together in Moscow, only the Moscow
  card may be transferred from one
  player to the other.) If either player
  holds more than 7 cards as the result
  of a transfer, the excess cards must
  be immediately discarded to the Player
  Discard Pile. 


Special for the Researcher:


  The Researcher may give a
  fellow player any card from his hand
  when involved in a Share Knowledge
  action. He is not restricted to giving
  the card of the current
  jointly-occupied city, like other
  players are. is freedom only applies
  when the Researcher is giving a
  card—he always receives a card with
  the same restriction as other players.
  e Researcher can utilize this ability
  when involved in Share Knowledge
  actions on any player’s turn.


It does NOT matter what player's turn it is. Either play can initiate the transfer and either give or take a card from the other player. The Researcher only gets to use his special ability of transferring any card when the card is being transferred from the Researcher to another player. In order to transfer a card from another player to the Researcher, they must be in the same city as depicted on the card being transferred. 

This will make the game a fair bit easier as you don't have to wait for the right players turn to transfer cards. You could play the other way if you wanted a bit more difficulty with 2 or 3 players, but playing it with 4 can be almost as bad as adding another epidemic card. :D