Yeah they get aware of the fact that there's an assassin in town, and they know what you look like too.
BTW if you haven't been, play the game over and TALK TO LUCY after each memory block... most of Desmond's story is gotten through those conversations, so you're missing a huge chunk of storyline if you ignore her after each memory block is complete and go to sleep instead.
You know when you can go to sleep when she says "aren't you tired?" followed by "can't talk.. gotta work".
XI Wiki

