Regular patronuses do not hide a person from dementors, so if a person came in with the true invisibility cloak, but not a true patronus, they would be noticed by the dementors as soon as they gave it to the escapee.
They would be in the same situation as a regular intruder, which are not all that rare, and whom the Dementors usually don't seem to immediately report about (otherwise the aurors would have wondered why that hadn't happened and have been much more alarmed).
Remember that the Deathly Hallows, in canon, were widely believed to be mythical. It's hardly fair to expect these people to even keep in mind that they exist, let alone contemplate exactly how powerful they might be and account for them in their plans.
In canon, but apparently not MoR. Santa Claus, Dumbledore and Snape all mentioned the true cloak to Harry right away.
Remember that when we, the readers, are encouraged to think of artifacts of great power in this setting, the Deathly Hallows immediately come to mind, but to a person who's grown up in the wizarding world, they're just three out of perhaps thousands of possibly-real objects out of myth and story.
Very likely not. If there were supposed to be thousands of comparable items in MoR rather than the dozen or so we know of the odds of most of the ones from canon being mentioned, but none of those extras would be pretty short even considering overlapping reasons for being mentioned.
In relation to population size true cloaks of invisibility are more common that jetpacks for us. If 20 FBI agents had two hours to think about an apparently impossible mystery that would be obviously solved if you assume a jet-pack I'd expect at least one of them to raise the possibility.
Although it's possible that the dementors would not quickly report an intruder who gave the true invisibility cloak to a prisoner, if they have that degree of reasoning ability, I suspect they would take more notice of a person leaving one of the cells who they hadn't seen coming in than someone entering.
Santa Claus gave him the true invisibility cloak knowing what it was, and Dumbledore had been searching for the deathly hallows for much of his life, and had good reason to suspect one was in Harry's possession. Snape is the odd one out, being the only on...
- This thread has run its course. You will find newer threads in the discussion section.
Another discussion thread - the fourth - has reached the (arbitrary?) 500 comments threshold, so it's time for a new thread for Eliezer Yudkowsky's widely-praised Harry Potter fanfic.
Most of the paratext and fan-made resources are listed on Mr. LessWrong's author page. There is also AdeleneDawner's collection of most of the previously-published Author's Notes.
Older threads: one, two, three, four. By tag.
Newer threads are in the Discussion section, starting from Part 6.
Spoiler policy as suggested by Unnamed and approved by Eliezer, me, and at least three other upmodders:
It would also be quite sensible and welcome to continue the practice of declaring at the top of your post which chapters you are about to discuss, especially for newly-published ones, so that people who haven't yet seen them can stop reading in time.