First of all, I agree. Often times they do mean "don't tell me anything about me that I don't accept". However, there is a reason why I believe many people don't actually understand the concept of judgement....
The way I see it, there are two types of judgment... there is internal judgment and external judgment (I made these terms up, I think)...
Internal judgment involves spiritual, emotional, and mental processes. It involves making decisions, comparisons, opinions, determinations and definitions... This, I believe is the type of judgement you are talking about... and as long as it is true and lines up with the Word of God, there is nothing wrong making those types of judgments.
External judgment involves administering consequences and taking matters into your own hands... For example, when God executes judgment... it's not just Him saying, "I think so and so", or "you're going to hell if you don't stop doing this" or , it's more like famine, disease, death, punishment, reward, etc... In this sense, I don't believe a saint has the right to do so, unless he/she is in a position of authority. This is the one I believe that the people you mentioned confuse internal judgment with.
God's requirements are clearly stated in the Bible... but only He has the right to administer the consequences as a result of disobeying them. For example, If you come to work everyday drunk, and I say you have an alcohol problem and offer help, that by definition would be internal judgement, there is nothing wrong with that because it's obvious that you show the signs of a person with an alcohol problem. But if I call you names like "whino", spread gossip about you and make your life a living hell, that constitutes external judgment, as I have now tried to step in God's place by punishing you and putting matters in my own hands.
So the way I see it, treating a sinner badly and rejecting him/her falls under the category of external judgment, which in this case we shouldn't do. We do have the right to our own opinions, and we do have the right to speak on what we believe is right as long as we do it in love and we do it in a manner that draws them to Christ, rather than rejecting them.
This is just the way I see it, it is by no means a guide to go by.