If you DEMAND PROOF every time you engage with someone it means you think he/she is lying. Engaging with liars makes no sense. Why do you?

I disagree. Asking for verification just means that you want to make sure the information is accurate. There is a difference between lying and just repeating misinformation. Well-meaning people can still get things wrong.  I recall you repeating a story about Trump not know Puerto Rico was a US territory. The only source for that I could find was an article that was attempting satire. I wouldn't call you a liar for posting that, but I would not repeat it without checking multiple sources to make sure it was true.