Most enlightened people don't even know they're enlightened.

The ones who arrived there without conscious effort, through some intense concentration on some other aspect of life, simply lose their angst. Asked to explain, they usually have some kind of muddled folk mysticism about how it happened. Very few of them teach.

I know of well attested enlightenment from an olympic-grade swimmer, and a guy who really liked fly fishing, for example. The swimmer counted breaths while swimming, very much like one particular zen practice.

The desire to go out and teach is much more a side-effect of the people that were trained in schools founded by people who decided to go out and teach: the desire to teach is not embedded in the enlightenment experience for most people.