Despite my analness with my own stories, I'm not a fact-checker in other people's stories either. I'm like you, Lore, where if I read something that intrigues me and makes me wonder if it's true or not, I'll look it up, but that's just to learn, not to check on the writer. I usually just go with the flow unless there are so many glaring errors that I can't maintain my suspension of belief. But in that case, I probably would have stopped reading already. Good writers are usually creative enough to make something seem believable even if there's no truth or reality to it at all.
As far as calling somebody out, I would never do that unless the author specifically asked. In that case, I'd be willing to help, but I don't like to be the one who's always pointing out mistakes in reviews. Contrary to what some may think, I don't want to be seen as a know-it-all LOL.
I have been called out by readers before. Sometimes it's helpful, and sometimes it's kind of annoying. When I was writing Broken, there was this girl who was a med student in another country who would constantly challenge me on stuff, and it drove me nuts because I knew I was doing the absolute best I could to get things right. Apparently I should have been running everything by her before I wrote it! I know she was just trying to help, and she was usually right, so what I learned from that is to fact-check before I write, while I write, and after I write and not be afraid to ask people with experience for help, instead of trying to look up everything on my own. At the end of the day, it's just fiction, so it doesn't matter if it's all 100% accurate or not, but if I can tell the story I want to tell and be 99% accurate at the same time, I'd like to be! That's just me being anal though.