It's really tough to answer your question without naming specific schools you are talking about. In general yes, even amongst elite schools, it can be easier at a private school to obtain certain opportunities and more intimate advising/support. When I was deciding a couple years ago, the common sentiment I heard most often from students at the publics I was looking at was, "nobody will hold your hand here" whereas students from privates never said this and gave off the vibe that they were better supported.
It is no coincidence that when you look at studies regarding student and graduate research productivity (there was a paper a couple of years ago titled "What makes a top medical school" that studied this), the top 10 schools were all private and I believe UCSF was the only one to break the top 20. This is despite the fact that schools like UCSF, UMich, UCLA are all top research institutions that get massive funding from NIH and recruit students who are interested/accomplished in research. From my experiences, elite privates do have a slight leg up when it comes to research opportunities trickling down to medical students.
That being said this is a very general concept, and it really depends on which specific schools you are looking at. Some private schools are more like publics, and some publics are more like privates.