You're right that lack of awareness and judgement is the problem.
Lack of awareness of the fundamental error of attribution, and rushing to judgement based on far too little information.
Anyone who thinks they can reasonably judge anyone based on any single post, much less 140 characters, is not doing a great job of being a citizen in the 21st century. Everything we say can be misunderstood, and if you can't imagine anyone tweeting something you don't like without them being racist/sexist/whateverist that is simply because your imagination sucks as a tool for knowing reality, just like mine, just like everyone else's.
What we can or cannot imagine has absolutely nothing to do with what is real. This is the lesson of three hundred years of science. This is why ideas must be publicly tested by systematic observation, controlled experiment and Bayesian inference to be counted as knowledge.
This is not to say we must never judge, but that judgement is far more difficult than most people realize, and they are far too quick to do it based on far too little information and far too much imagination. [edit: I should have said "we are far too quick..." The habit of judgement is hard to break!]