At schools that don't meet full need, how do merit and need based aid interact?

It does not necessarily mean that any given admit will be offered that percentage of need met. Of course, the college can define “need” however it wants, so percentage of need met is does not really say that much anyway.

Examples of some different college policies on outside scholarships from a few years ago (may have changed): List of outside scholarship + financial aid policies ("stacking", "scholarship displacement")

Rose-Hulman does not say on its web site what happens when a student with need-based financial aid gets merit scholarships.