If you go to a character’s profile and click on the “View all Friends” button, the screen that pops up purports to tell you your mutual friends. However, it doesn’t work right. It shows you all of your own friends in the “mutual friends” area, regardless of whether the character you’d been viewing is friends with them.
This is a much bigger deal in 0.4 than in previous versions, since you can no longer invite random strangers to events that they would like (they’ll now refuse if you have no mutual friends, or something like that anyway).
Also somewhat related: The “Mutual Friend Preview” on other characters profiles just shows random friends of the person, who may have no connection to you.