I found one example where she notices Harry sneaking out. When he goes to find the Mirror of Erised in Philosopher’s Stone, she seems a bit surprised:
“Who’s there?” squawked the Fat Lady. Harry said nothing. He walked quickly down the corridor.
But we never see any evidence that a teacher (Dumbledore aside) was aware that somebody might have been wandering the castle at night.
Remember that invisibility cloaks are fairly rare, and it is fairly unlikely that a student would have one. At the very least, she won't be expecting it. There's no real evidence that anybody has left the tower (if Harry was careful with his footsteps.) So what would she say? The portrait hole opened, somebody went out, but I didn't see or hear anything. Great story.
I’m sure she can think of plausible explanations for these events that don’t involve super-rare invisibility cloaks. Perhaps it was a first-year playing a prank. Perhaps somebody was about to come out, but remembered something they’d left inside. And so on.
That’s the charitable interpretation.
The other interpretation is that she doesn’t notice or care, because she’s just not very good at her job. She gets drunk or falls asleep on the job, gets tetchy with students, and will refuse to open if she’s not in a good mood. Hardly the hallmarks of a good security guard.
