People, more often than not, confuse eternal judgement (not ours to hand out) and the common idea of judgement (done every day by almost everyone). That's why they carelessly say, "Don't judge me." when in fact we should.
Amen! A lot of people, even good intentioned Christians, are confused about this topic.
Exactly, what people usually mean by "don't judge me!" means that they want you to ignore the blatant, destructive sins they are living in.
Obviously, pointing those things out in public about an individual isn't how we go about it, but pointing out that such things are wrong when the society is calling them good is something that we should be doing. If society says homosexuality is right, we need to stand out and say it is wrong. We aren't judging anyone's souls or salvation by doing so. Yet some other people accuse us of getting caught up in the "culture war" and that we should shut up and keep our religion in the private sphere. Clearly they don't know what Christianity has always done...