The plot twist at the end of Prisoner of Azkaban is that Sirius Black is in fact a secret Animagus, the mysterious black shaggy dog that Harry has been seeing all year. Harry doesn't even contemplate that the dog may be an Animagus because he thinks it's the Grim. However, there were two people at Hogwarts who did know Sirius' secret and indeed had an interest in catching him.
Both Remus Lupin and Severus Snape believe Sirius to be guilty of thirteen murders at this stage. Neither of them believe in his innocence. Both of them cared deeply about protecting Harry. Both of them believed that Harry was Sirius' number one target. Both of them believed that he was breaking into Hogwarts to try and kill Harry. And both of them, as upstanding wizarding citizens, had a more general interest in catching a dangerous fugitive. Snape also has the added incentive that he hates Sirius' guts.
So they would both have handed in Sirius if they'd known where he was. And they both knew what Sirius looked like in his Animagus form and would've instantly recognised him. Yet Sirius made numerous appearances that year around the Hogwarts grounds. Namely:
He appeared at the Quidditch Stadium in front of the whole school.
...at that moment a flash of lightning illuminated the stands, and Harry saw something that distracted him completely: the silhouette of an enormous shaggy black dog, clearly imprinted against the sky, motionless in the topmost, empty row of seats.
(Prisoner of Azkaban, Chapter 9, Grim Defeat).
Harry thought he saw him on the way back from training one time.
They were half-way towards the castle when Harry, glancing to his left, saw something that made his heart turn over - a pair of eyes, gleaming out of the darkness...He took a deep breath as relief seeped through him; he had been sure for a moment that those eyes had belonged to the Grim.
(Prisoner of Azkaban, Chapter 13, Gryffindor Versus Ravenclaw).
Harry definitely does see him on the edge of the Forbidden Forest with Crookshanks.
And next moment, it had emerged: a gigantic, shaggy black dog, moving stealthily across the lawn, Crookshanks trotting at its side. Harry stared. What did this mean? If Crookshanks could see the dog as well, how could it be an omen of Harry's death?
(Prisoner of Azkaban, Chapter 15, The Quidditch Final).
Finally, he attacks Ron/Wormtail.
But before they could cover themselves again, before they could even catch their breath, they heard the soft pounding of gigantic paws. Something was bounding towards them out of the dark - an enormous, pale-eyed, jet-black dog.
(Prisoner of Azkaban, Chapter 17, Cat, Rat and Dog).
Harry, a random student, sees Sirius in dog form four times at Hogwarts (not including the time he saw him in Surrey after blowing up Marge). How did Snape or Lupin not see Sirius the entire school year? He seems to be roaming the Forbidden Forest pretty openly the whole time and even turns up to a Quidditch match.
In addition, Lupin had the Marauder's Map for the second half of the year. He eventually saw Wormtail on the Map through a random chance encounter. Why didn't he notice Sirius on the Map?
Is there any explanation for how Sirius managed to hide himself from Lupin and Snape for a whole year despite the fact that they knew his secret disguise?