I'm a librarian who loves anime, manga, and reading a wide variety of genres.
Leslie asks Peter whether he should be telling her all this stuff about magic and magical beings and Peter says "I don't see why not. Nightingale's never told me not to."
Except here's the thing: I'm pretty sure it was part of Peter's oath before becoming Nightingale's apprentice that he wasn't supposed to tell anyone about this stuff. It was a little vague - I interpreted it as "don't tell anyone who isn't part of our secret magical club," although I suppose it could have been interpreted as "don't tell anyone who isn't also part of the police." Either way, I don't think Peter should have assumed that he was allowed to tell Leslie all of this.
It's a moot point, since Nightingale never gets upset with Peter about this, but it's a detail that bugs me.