Here is an example of how Stanford handles outside scholarships:
https://financialaid.stanford.edu/aid/outside/
Here is a page where UCSD describes how scholarship money is applied for those students receiving need-based financial aid (scroll down to “FACTS ABOUT SCHOLARSHIPS”):
http://fas.ucsd.edu/types/scholarships/index.html
Obviously, each college may have its own policy on how scholarship money interacts with need-based financial aid. For any specific college, look around its web site, or ask directly.