Quirrell is reading Hermione and Draco's minds, and the story he told Harry about how he learned that Harry was a parselmouth, in chapter 49, is partially fabricated. While that story does explain Quirrel knowing that Harry's a parselmouth, it doesn't explain why he chose to confirm that knowledge on his very next private meeting after Draco found out. Also, as Harry observed, Quirrell has at least one hidden source of information:
(EDIT: On rereading, Harry brought up parselmouth first, which explains the timing. But the remaining arguments for the conclusion that Quirrell is reading Hermione and Draco's minds still apply, and still seem sufficient.)
"There were times when Harry suspected that Professor Quirrell had way more background information than he was telling, his priors were simply too good."
And besides that, simply as a prior probability, Quirrell ought to be reading every mind he's confident he can get away with reading, and Hermione and Draco are very unlikely to notice . This also suggests that when Quirrell arranged for Harry to learn occlumancy, it was a bit of misdirection; he knew he'd be able to get the same information from Harry's friends, but that having suggested it would make Harry more inclined to trust him. Finally, this means that the secret of partial transfiguration is not safe, and if Quirrell is Voldemort then it does not satisfy the conditions of the prophecy.
it was a bit of misdirection; he knew he'd be able to get the same information from Harry's friends,
Harry's friends? Harry is a compulsive secret keeper! He will not even tell Hermione how to make patronuses and he keeps what he does with Malfoy and Hermione a hidden from each other (and incidentally manipulates them by keeping the key details of what he is trying to do to them to himself.)
[Update: and now there's a fifth discussion thread, which you should probably use in preference to this one. Later update: and a sixth -- in the discussion section, which is where these threads are living for now on. Also: tag for HP threads in the main section, and tag for HP threads in the discussion section.]
The third discussion thread is above 500 comments now, just like the others, so it's time for a new one. Predecessors: one, two, three. For anyone who's been on Mars and doesn't know what this is about: it's Eliezer's remarkable Harry Potter fanfic.
Spoiler warning and helpful suggestion (copied from those in the earlier threads):
Spoiler Warning: this thread contains unrot13'd spoilers for Harry Potter and the Methods of Rationality up to the current chapter and for the original Harry Potter series. Please continue to use rot13 for spoilers to other works of fiction, or if you have insider knowledge of future chapters of Harry Potter and the Methods of Rationality.
A suggestion: mention at the top of your comment which chapter you're commenting on, or what chapter you're up to, so that people can understand the context of your comment even after more chapters have been posted. This can also help people avoid reading spoilers for a new chapter before they realize that there is a new chapter.