I believe Hogwarts was generally disconnected from the Floo network, although connections could be made if necessary (or by force).
In particular, Floo connections directly to student common rooms would be problematic from a child protection aspect. Some members of staff might have theirs connected, but I don’t recall instances when staff used them. In general, I can’t think of many instances when the staff leave Hogwarts during term (Dumbledore aside, and he has methods other than Floo).
In Half-Blood Prince, students return to the school by Floo at Christmas, but this is described as a special case.
Late in the afternoon, a few days after New Year, Harry, Ron, and Ginny lined up beside the kitchen fire to return to Hogwarts. The Ministry had arranged this one-off connection to the Floo Network to return students quickly and safely to the school.
Pottermore adds weight to this theory (in spoiler tags if you haven’t got to this part):
The fireplaces of Hogwarts are not generally connected, although there have been occasions when one or more has been tampered with, often without the staff’s knowledge.
In both instances where the Floo network is used within Hogwarts, it’s only used for a person’s head/as a communication tool, not for full bodily transfer. I would guess that the security measures prevent you entering or leaving the grounds via Floo, just like apparition (special exceptions aside). The relevant passages are quoted below.
Undoubtedly, head-only transport isn’t as much of a risk as, say, allowing Death Eaters direct physical passage into the school, but it’s still a security hole. When Amos Diggory uses Floo to reach the Weasley’s, we see that this head-only transport still allows physical objects to be transferred:
Mrs. Weasley took a piece of buttered toast from a stack on the kitchen table, put it into the fire tongs, and transferred it into Mr. Diggory’s mouth. “Fanks,” he said in a muffled voice, and then, with a small pop, vanished.
Whether this is an intentional omission or an oversight, I don’t believe is addressed in the canon.
Uses of the Floo network in GoF/OotP
In Goblet of Fire:
Harry looked back into the flames, and jumped. Sirius’s head was sitting in the fire. If Harry hadn’t seen Mr. Diggory do exactly this back in the Weasleys’ kitchen, it would have scared him out of his wits.
There’s no evidence that Harry or Hermione tampered with the Gryffindor fireplace to allow Sirius to Floo in that evening, but clearly he was able to. This is an apparent hole in the Floo network security.
We can use this to surmise that Hogwarts Floo traffic is not monitored, as there is never any indication that staff or Ministry officials were aware of Sirius’s presence. If they were, it could probably have been used to trace his location.
In Order of the Phoenix, when Harry uses the Floo network without detection:
It was one of the most curious sensations he had ever experienced. He had traveled by Floo powder before, of course, but then it had been his entire body that had spun around and around in the flames through the network of wizarding fireplaces that stretched over the country. This time, his knees remained firm upon the cold floor of Umbridge’s office, and only his head hurtled through the emerald fire…
As far as we know, Umbridge never knows about this conversation. (When she catches Harry the second time, it’s because of Stealth Sensoring Spells used to prevent Niffler attacks, not a notification from the Floo authority.)
Sirius and Dumbledore are both writing to each other during Goblet of Fire:
“You are not Sirius’s only correspondent,” said Dumbledore. “I have also been in contact with him ever since he left Hogwarts last year. It was I who suggested the mountainside cave as the safest place for him to stay.”
If Dumbledore knew that Sirius wanted to speak to Harry, he could have suggested or supported using the fireplace in the Gryffindor common room, and opened access to the Floo network for those specific instances. Otherwise, the network could be completely shut at all times.
Since I don’t have canon information about whether this theory is true, I’m marking it as speculation.