united states - Impact of type of institution (public vs. private) on long-term tenure track career perspectives

I am curious about to what degree and how, if any, type of educational institution (public vs. private) has an impact on long-term career perspectives in academia (with eventual target of a tenure-track faculty member). For the purposes of this question, please assume we are talking mostly about educational institutions in the U.S. with a very high research activity per Carnegie Classification.