What would happen when Harry and Hermione would meet other Harry and Hermione while using time travel clock? For example scene of execution of hippogriff in Prisoner of Azkaban.
I'd say that one of them (not time-traveling yet) would be pretty shocked. Even Hermione said it before they ran to save the hippogriff. Hermione would probably be okay, because she already knows that she might meet herself in the future.
Harry saw himself saving himself (yes, that's really paradox) and the only effect was a confused Harry, because he thought it's his killed dad.
You guys are, of course, missing the simple point that since future Harry has already LIVED through the past situation, he is armed with the knowledge that past Harry never ran into future Harry visiting (otherwise future Harry would have the memory of having run into someone who looked exactly like him). Everything that happens when they travel to the past is exactly what had ALREADY happened by the time they made the decision to go to the past in the first place. In other words, they couldn't screw anything up by 'accidentally' running into themselves, because if it had happened, they would already know about it.
The only reason Harry didn't have the memory of seeing himself from the future is because it was too far away for him to see the figure properly, and he assumed it was his father, until he went back in time, saw his past self across the lake and then realised that he was currently the figure past Harry thought was his father.
Hermione does mention that time-travelling wizards have been injured or even killed by their confused previous selves. Understandable if you didn't know time-travel was possible and suddenly come across somebody posing as yourself in a world of magic and polyjuice potions (though, I'd think about stunning first and asking questions later)
Harry thought he was seeing his father in the night-dark forest, so while confused, didn't feel threatened (plus, he had enough trouble with the Dementors swooping in).
Due to the fact that the non-traveling Hermione already knows about the watch, she would pull non-traveling Harry away and maybe explain the watch to him. It could become difficult when Harry rescues himself with the Patronus; because he knows he can do it he won't have such strong feelings as if he is in fear.
In the books, Hermione addresses this in a little more detail than in the movie, but the movie does explain it a bit. Basically, there's no paradox preventing someone from visiting their past or future self, but...
if Harry were to burst in on the three people in Hagrid's hut, including his past self, his past self might think he'd gone mad, or that his future self was some trick or illusion, and Harry's future self might be attacked by any of the people in the hut. Even if his past self believed that his future self was really him, changing the course of time so radically might have unintended consequences. If Harry had killed Pettigrew alias Scabbers in the hut, then Scabbers wouldn't have run off toward the tree, and all the events that unfolded, including Harry learning that Sirius was innocent and thus wanting to go back in time to save him, would never have happened. By remaining unseen, they changed time in such a way as to preserve the basic timeline and thus all knowledge of the events, thus not removing the reason they would have gone back in time in the first place.