I can correct my brother's spelling/grammar/factual mistakes in public and he's cool with that (although he doesn't always correct me in public). When I was a kid, I was even told that I am allowed to correct my teachers if/when they're wrong.
Having grown up with people like that, I took the liberty to correct people when I thought they were wrong. Not many people took it well, especially when they were not close friends with me. I learned from my experiences and now I don't always say it when I find something wrong. I mostly assume that they don't care if they are right or wrong; they just want to say what they think/feel.
But I really wish everyone were like my brother. I also wish everyone told me when they thought I was wrong.