A special thanks goes to my friend Marina for this. Go to North Clocktown and cut the circle of bushes. When they come alive and start to run away, put on the Keaton Mask. Unless you waited too long (a rare occurrence), the obnoxiously cute Keaton will spring up from the ground and ask you some trivia questions. (The answers can be found on this website in the information section.) Get them right the first time and he'll give you a Piece of Heart; Do it any other time and you get a red rupee.
Mask of Kafei
The Curiosity Shop Owner is an old friend of Kafei's.
When wearing any other mask he'll refuse service to you,
but when he sees you wearing Kafei's mask, he'll begin to
stutter. The Postman also acts suspiciously towards the
mask, but he's much more calm and composed. Speaking to Anju while
wearing it starts up the sidequest. (I have also provided
the complete walkthrough for this sidequest in the information
section.)
Sakon is too busy eyeing your sword to pay attention to anything else, so don't even bother. The Happy Mask Man also seems rather uninterested, saying only that it comes from a worried mother. This helps to prove that Kafei probably didn't steal the mask, since the mask-lover has a BIT of a temper on him.
The sign-reading carpenter is either Anju's admirer or her father, because gets pretty angry when you speak to him with the mask on. He says to stay away from "his Anju", and considering how old he looks, he's more likely her dad. Anju's mother also gets rather cranky: visit her at the Stock Pot Inn after her daughter's shift is over, and she'll tell you not to wear "such an unpleasant mask". Anju's grandma is kind of senile and will always call Link "Tortus" (Anju's father's name), no matter what mask he wears.
The other responses are somewhat commonplace. Most people will say they haven't seen him, and The Bomber Kids spill the beans about Kafei's location. (It's not like you didn't already know, but still...) They're also curious as to why he's gone missing, the nosy little suckers!
Without wearing a mask...?
Several people mention Anju and/or Kafei without Kafei's Mask being
equipped. Cremia is apparently flustered about her friend
Anju getting married when the moon is expected to fall if you take
the wagon ride with her. (By the way, putting the Circus Leader's
Mask on before taking the wagon ride helps greatly...just
thought I'd add to the spoiler-happiness of this website.)
When in Sakon's Hideout, Chat complains about not liking Kafei. It's obviously because he told Link to step on the switch, and besides, Chat's not a bed of roses herself.