Students must meet the sponsor's criteria in order to receive a private scholarship. The criteria may or may not include a student's financial need.
Private scholarships can be very competitive, i.e. many applicants for a few awards.
Most financial aid is based on need. Many colleges also offer scholarships based upon other factors such as academic achievement or special talents. They generally will require students interested in these funds to complete the financial aid process anyway to make sure they receive all aid to which they are entitled. You definitely should apply for everything for which you are eligible and that interests you. If you receive need-based financial aid from a college and a private scholarship, you will be required by your college to inform them of the scholarship amount.