Schools used to do this, but the push for risk reduction, metrics, and rules has become so great that it no longer happens. There used to be thousands of student-run organizations in schools across the country. That wasn't a euphemism like it is now-there were no teachers or other adults involved. The kids running them did have to learn to cooperate and make decisions in a responsible way, or face the natural consequences of the group falling apart and social failure.

Now, such organizations are banned. The closest analogue is a "student" council, run by an adult, that might get to choose the color of the wallpaper at prom.