The friendship level has to be very high in order for them to send you their pic after leaving. (i.e. they write you letters with presents constantly, they give you furniture and clothing at random times, the invite you to their house or come to yours unexpectedly a lot)
What bothers me more is that once a villager is gone, everyone else has total amnesia about the villager that left. Before the villager moves out, all the rest of them are so concerned that so-and-so is moving. His or her potential departure is the central piece of gossip in town for the five days leading up to the move. Then the villager moves, the house plot is gone, and none of the other villager ever says another word about them ever again. I find it very creepy.