We always draw characters at random, and until I read this post, it hadn't occurred to me that we have yet to run the assassin in any of our games.
I like Old Master's idea that assassins can only use their special powers when a monster is already face-up on the board. It makes cognitive sense. In effect, when you draw a monster from the adventure deck, you are being attacked by the monster: You've accidentally wandered into the Dragon's hunting ground, for instance, and the dragon has decided to make you his lunch.
On the other hand, if the Dragon is already on the board, then you've chosen to enter the Dragon's hunting ground fully aware that the dragon is in the area. Thus, you are the aggressor/attacker. You could therefore try to sneak up on the sleeping dragon and thrust your sword into it's heart before it has a chance to respond.
Rule of Thumb: Any time your character (Character A) moves into a space occupied by a monster or another character (Character B), then your character (A) is the ATTACKER. Any time a monster or other character (B) moves into, or is placed upon, a space that your character (Character A) has already entered, your character (A) is the DEFENDER.
Since a standard turn requires you to move first, and then draw an adventure card, a monster on the adventure card is the ATTACKER and you are the DEFENDER.
( You might rule that objects or powers that let you see the next adventure card would give you the element of surprise.)