As repetition,
it's going to vary from school to school. I'm going to guess that you'll see more scholarships, grants, and tuition reductions offered to those who attend a private institution as opposed to a public institution (which are already heavily subsidized to begin with).
When you file FAFSA for med school, nearly everyone is going to have a EFC of 0, because you are filing as an independent. The exceptions are those who have had other careers beforehand or have spouses. This EFC basically means nothing, it just shows you eligibility for Sub and UnSub Stafford loans and also GRAD-plus loans. Everyone gets these.
While filing as an independent, you don't HAVE to submit your parental information, but don't expect any need-based scholarships or grants from the school. With nearly everyone having an EFC 0, the parental information is the only way to differentiate between students.
And again, it's going to vary from school to school. My parents make <$45,000 gross /year. But my school doesn't give out many scholarships (merit-based or need-based), plus the tuition is already very cheap in comparison to many other medical schools. As such, I didn't get any grants, scholarships, etc.
Each school you interview at should give you a talk about financial aid. This is when you should ask about the availability of scholarships, grants, etc.