I'll start out with these. They're from Harry Potter and the Philosopher's Stone. People are free to recommend new entries. Or I might end up editing this to be a community wiki. Hunting for this type of thing might warrant some collaboration.
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"I'd love to have a dragon; vastly misunderstood beasts, they are." [in front of a Muggle passenger. this is a deleted scene.]
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"... and then I lent him to Dumbledore to guard the—"
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"what [Fluffy] is guarding is between him [Dumbledore] and Nicholas Flamel."
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"Ain't no one gonna get past Fluffy. Not a soul knows how, except for me and Dumbledore."
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"Take Fluffy, for example, just play him a bit o’ music, and he falls straight to sleep."
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"Yes!" [loudly cheering that Gryffindor won the House Cup when faculty and staff are supposed to be unprejudiced; he promptly shifts to quiet, respectful clapping when his self-awareness kicks in. the information he divulges here is about himself and his bias for Harry, Hermione, and Ron possibly in front of teachers.]
Hagrid seems to become more cognizant as the series progresses, but I'd need to re-watch all of them to verify if there's other instances. He also has less screen time after the first few films.