Author-requested missions are not actually a very rare phenomenon. With news of the PPC spreading across the Internet and many serious fanfiction writers having grown out of their Mary Sue phases, there are more than a few people with the desire to teach others how not to write a Mary Sue.
Enter the PPC. Sometimes it's a Boarder or a friend of a Boarder who requests the mission; sometimes it's an author who agrees with constructive criticism. The fic, if judged to indeed be badfic, enters the system like any other and is assigned to agents—occasionally, to the same agent who wrote the fic.
The author of the fic may or may not help the PPC in their task.
Author-requested missions differ from trolls in that the fics in question are honest attempts made when one was (most likely) younger and (almost certainly) sillier. Trolls are simply deliberate badfic made to get attention through being outrageously bad or otherwise offensive.