Dementors are probably of human intelligence
People generally treat Dementors as if they were capable of formulating complex thoughts. For example, when the Dementors search the Hogwarts express, Lupin has this to say:
"And Professor Lupin stepped over you, and walked toward the dementor,
and pulled out his wand," said Hermione, "and he said, 'None of us is
hiding Sirius Black under our cloaks. Go.' But the dementor didn't
move, so Lupin muttered something, and a silvery thing shot out of his
wand at it, and it turned around and sort of glided away.... "
—Harry Potter and the Prisoner of Azkaban
Lupin seems to believe that not only can the Dementor comprehend what he is saying, but it understands the relatively complex idea that they are not concealing Sirius Black.
Dumbledore does imply that they have a very alien nature, but in doing so affirms their intelligence:
"It is not in the nature of a dementor to understand pleading or
excuses. I therefore warn each and every one of you to give them no
reason to harm you.
—Harry Potter and the Prisoner of Azkaban
Not"Dementors are incapable of understanding you" but "It is not in their nature to do so."
Sirius speaks of the Dementors as believing that he is going insane:
Dementors can't see, you know...." He swallowed. "They feel their way
toward people by feeding off their emotions.... They could tell that
my feelings were less -- less human, less complex when I was a dog...
but they thought, of course, that I was losing my mind like everyone
else in there, so it didn't trouble them.
—Harry Potter and the Prisoner of Azkaban
Sirius doesn't even seem to be speculating, so perhaps the Dementors actually talked about this among themselves somehow.
Snape seems to think the Dementors will be pleased to have Sirius back in their custody:
All I have to do is call the dementors once we get out of the Willow.
They'll be very pleased to see you, Black... pleased enough to give
you a little kiss, I daresay...."
And while Snape may have been exaggerating, a Dementor certainly seemed to recognize Barty Crouch Junior, and sucked out his soul:
But Professor McGonagall’s voice drowned Fudge’s. “The moment that —
that thing entered the room,” she screamed, pointing at Fudge,
trembling all over, “it swooped down on Crouch and — and —”
—Harry Potter and the Goblet of Fire
Note that not only was this not what Fudge had ordered it to do (though he tried to justify it after the fact), but the Dementor did not attack anyone else as it passed through the castle. It personally remembered Crouch and wanted to Kiss him—"the moment" it "saw" him.
The Ministry can order Dementors to do rather specific things—for example, to attack Harry Potter.
“Yes, you have,” said Fudge forcefully, “and I have no reason to
believe that your views are anything other than bilge, Dumbledore. The
dementors remain in place in Azkaban and are doing everything we ask
them to.”
“Then,” said Dumbledore, quietly but clearly, “we must ask ourselves
why somebody within the Ministry ordered a pair of dementors into that
alleyway on the second of August.”
—Harry Potter and the Order of the Phoenix
Most telling of all, of course, is that Voldemort was able to sway their loyalty away from the Ministry by offering them a more bountiful supply of prey, also implying some sort of intelligence.
“It is with almost equal regret that we report the mass revolt of the
dementors of Azkaban, who have shown themselves averse to continuing
in the Ministry’s employ. We believe that the dementors are currently
taking direction from Lord — Thingy."
—Harry Potter and the Order of the Phoenix
Dementors clearly have a fundamentally nonhuman mindset. As Hagrid said, they care nothing for guilt or innocence as long as they can suck the happiness from people. As Dumbledore said, they do not accept excuses or justifications. Their allegiance is to whomever can offer them the most prey. The quote given by the questioner indicates that Dementors are vicious: they care little for whom they harm. But they are fairly consistently treated as having an high level of intelligence.
Dementors are not listed as beings in Fantastic Beasts and Where to Find Them. However, this does not indicate that they do not possess sufficient intellect or other qualities to qualify as "beings," since there are many other creatures that are manifestly beings and are not mentioned in Fantastic Beasts and Where to Find Them, e.g. Veela and giants. It would seem that one possible criterion is "requesting status as beings," as mentioned in the entry for Leprechauns, something that Dementors would likely be supremely disinterested in.
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