I can't help thinking that the Twitter interface is a contributing factor. First is the ease in messing up the direct message syntax. Mistype a single character and your tweet goes awry. Second, if your tweet is too long, and your provider breaks the text up into multiple messages, only the first message remains a direct message; the rest of the tweet, quite reasonably enough, goes out publicly because it's not preceded by a direct message flag.
I don't have an easy fix for this. What I do is to use Twitter solely from the web interface or from a secondary handset on a separate number from my main phone which is generally my emergency phone. This means I have either a tangible reminder of the fact that texts from a given phone can go out to Twitter on the hand, or I'm actually looking at a Twitter interface on the other.
I'm not suggesting this is something everyone should do, but it's noteworthy that my second phone is a cheap prepaid handset. I think prominent people might be well-advised (by their publicists or PR people) to spring for something like that.
Or they could just not care and continue to provide everyone with the best in schadenfreude entertainment.