In other words, would a directional antenna used as a receiver be more sensitive to signals coming from one direction, or be equally sensitive to signals from all directions?
As Peter's answer says, the short answer is "yes".
The reason is, according to Wikipedia,
It is a fundamental property of antennas that the receiving pattern (sensitivity as a function of direction) of an antenna when used for receiving is identical to the far-field radiation pattern of the antenna when used for transmitting.
This is one consequence of the reciprocity theorem of electromagnetics. The proof is kind of hairy, though.