"To do something on somebody" is used when somebody does something undesirable in your presence. At least that's how I've seen it used most of the time.
Don't you dare die on me!
Don't get sick on me now!
I agree that "on me" adds a specific coloring. I would say that it indicates that it suggests that what happened was to the person's disadvantage. Here is the relevant definition of on in Dictionary.com:
21. Informal . so as to disturb or affect adversely: My hair dryer broke on me.
"He disappeared on me" suggests that his disappearance caused you problems of some kind.