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