Sirius had already been arrested by the time Hagrid delivered Harry to Privet Drive.
Let us reconstruct the timeline of events:
In Prisoner of Azkaban Chapter Nineteen Sirius tells Harry the following:
The night they died, I'd arranged to check on Peter, make sure he was still safe, but when I arrived at his hiding place, he'd gone. Yet there was no sign of a struggle. It didn't feel right. I was scared. I set out for your parents house straight away. And when I saw their house, destroyed, and their bodies... I realized what Peter must have done... what I'd done....
This establishes that Sirius went to Godric's Hollow on the night of October 31st (the night the Potters were killed). Now let's turn to what Hagrid says in Chapter One of Philosopher's Stone:
"Borrowed it, Professor Dumbledore, sir," said the giant, climbing carefully off the motorcycle as he spoke. "Young Sirius Black lent it to me. I've got him, sir."
This tells us that Hagrid hat met Sirius at some point before arriving at Privet Drive, but it doesn't tell us when that point was.
In Prisoner of Azkaban Chapter Ten Hagrid gives us the additional crucial detail:
Jus' got him outta the ruins, poor little thing, with a great slash across his forehead, an' his parents dead... an' Sirius Black turns up, on that flyin' motorbike he used ter ride.
This tells us that Hagrid's meeting with Sirius also occurred on the night of October 31st.
After Hagrid tells his story, we find out when Sirius was caught:
A long silence followed Hagrid's story. Then Madam Rosmerta said with some satisfaction, "But he didn't manage to disappear, did he? The Ministry of Magic caught up with him next day!"
"Alas, if only we had," said Fudge bitterly. "It was not we who found him. It was little Peter Pettigrew — another one of the Potters' friends. Maddened by grief, no doubt, and knowing that Black had been the Potters' Secret-Keeper, he went after Black himself."
Fudge only contradicts the assertion that it was the Ministry that found Black, not the assertion that it was the next day. Thus, we can assume that Sirius was caught the next day, which would be sometime on November 1st.
Going back to Philosopher's Stone, we know that the dropoff at Privet Drive did not occur until November 1st, because throughout the first chapter we see (from Vernon Dursley's perspective) wizards celebrating throughout the day; i.e. Voldemort had been defeated the previous night. When Dumbledore finally arrives, we have the following description:
In fact it was nearly midnight before the cat moved at all.
So Hagrid borrowed Sirius's bike on the night of October 31st, Sirius was apprehended at some time on November 1st, and the meeting at Privet Drive began slightly before midnight on November 1st and it was perhaps already the early moments of November 2nd by the time Hagrid arrived. So by the time Hagrid was leaving Privet Drive, Sirius had already been apprehended. In fact, for all we know, he may have already been in Azkaban, since as we know from Goblet of Fire Chapter Twenty-Seven he was sent to Azkaban without a trial:
"Oh I know Crouch all right," he said quietly. "He was the one who gave the order for me to be sent to Azkaban — without a trial."
This being the case, there was no way that Hagrid would be able to meet up with Sirius again after leaving Privet Drive. Presumably, Dumbledore was aware of this; an incident of such proportions might have been already public knowledge even without taking into account the fact that a man of Dumbledore's influence and power likely would have been informed immediately. Additionally, we know Dumbledore was at least somewhat involved because in Prisoner of Azkaban Chapter Twenty-One he said that he gave testimony:
I myself gave evidence to the Ministry that Sirius had been the Potters' Secret-Keeper."
So there was really no need for Dumbledore to warn Hagrid and alert the Ministry. The Ministry already had Sirius in custody, and Hagrid might have already known this as well. As for why Hagrid would say that he's going to return the bike to Sirius, and why Dumbledore would not have at least told him that it would be a waste of time, I think we have to look at what Hagrid really said. Here is the quote:
"Yeah," said Hagrid in a very muffled voice, "I'd best get this bike away. G'night, Professor McGonagall — Professor Dumbledore, sir."
He doesn't specifically say that he's going to return it to Sirius. All he says is that he's going to get it away. Now a reader at that point would have no reason to suspect that Hagrid meant something other than "return it to Sirius", in hindsight it can certainly be interpreted more broadly. This is especially so, given that in Prisoner of Azkaban Chapter Ten, Hagrid says that Sirius told him he could keep the bike:
Told me ter take his motorbike ter get Harry there. 'I won't need it anymore,' he says.
Though the use of the terms "borrowed" and "lent" in Philosopher's Stone might imply otherwise, it is possible that Hagrid was not being perfectly precise.
(It would still be somewhat odd that they talk about Sirius so matter-of-factly, without anyone mentioning that he had been arrested mere hours earlier.)