I just began a reread and it seems odd that Prof. McGonagall was at Privet Drive--all day watching the Dursleys--yet did not seem to know that Dumbledore planned to leave Harry with the Dursleys. Indeed, she seems almost shocked when Dumbledore suggests he is leaving Harry there. If she was surprised at this, why would she be at 4 Privet Drive in the first place? Did someone tell her to be at that particular place? Did she know Petunia? Is there any canon-based explanation for this, and if not, any other explanations? Thanks.
There's a simple answer to this.
"Hagrid's late. I suppose it was he who told you I'd be here, by the way?"
Mcgonagall earlier expressed surprise that Lily and James were truly dead. Obviously, she had no idea who the Dursleys were, she was just watching their news:
"even the Muggles have noticed something's going on. It was on their news."
She was told that she could find Dumbledore there after all the rumors by Hagrid. That's why she was there.
She was there all day because...well, we're not sure why. She does express her discontent for the celebrations, when there's danger around:
"Oh yes, everyone's celebrating, all right," she said impatiently. "You'd think they'd be a bit more careful, but no -- "
She waited the day for Dumbledore, but did not know that he would leave Harry there. It's all in the chapter.
She wanted to find out from Dumbledore whether Lily and James Potter were truly dead.
CHEESE covers most of the points but this passage makes the answer super-clear.
Professor McGonagall shot a sharp look at Dumbledore and said, "The owls are nothing to the rumours that are flying around. You know what everyone's saying? About why he's disappeared? About what finally stopped him?"
It seemed that Professor McGonagall had reached the point she was most anxious to discuss, the real reason she had been waiting on a cold hard wall all day, for neither as a cat or as a woman had she fixed Dumbledore with such a piercing stare as she did now. It was plain that whatever 'everyone' was saying, she was not going to believe it until Dumbledore told her it was true.
"What they're saying," she pressed on, "is that last night Voldemort turned up in Godric's Hollow. He went to find the Potters. The rumour is that Lily and James Potter are - are - that they're - dead."
Dumbledore bowed his head. Professor McGonagall gasped.
(Philosopher's Stone, Chapter 1, The Boy Who Lived).
She has come to Privet Drive because that was the only place she knew where to find Dumbledore. She would only believe the rumours about Voldemort/the Potters when she heard them confirmed by Dumbledore in person. Hagrid told her that Dumbledore would be coming to Privet Drive but not when. So she has no choice but to sit on a cold stone wall all day.
Bit of a dick move from Hagrid when you think about it really.