[noobhacker]

That's the opposite of my experience and understanding as a fresh PhD. In my field, social science, PhD students 100% own their research agenda. We get paid via a combination of grant (i.e. free money) and RA / TA work (i.e. money you work for). The better the program, the more of the money is free. And you always have the option to apply for external grants to fund yourself. In my (very fortunate) case, I rarely work on others' projects for more than 5 hours a week.

My friends in the natural sciences are more tied to the projects of their supervisors. But in the end they are still expected to be a mature researcher with their own agenda.

Caveat: I'm only certain that this is true for top-50 programs.