Yes, you can! If a player has a villager packed up and ready to move out, you can visit their town and ask that villager to move into yours as long as you have an open space in your town.
I've owned this game for a few years, but I haven't done too much multiplayer so I'm not sure. I've been trying to get Kiki for a long time now in NL, is there any way I could get her from someone?