We consider a model in which schools and colleges compete for high-ability students, which are independently identified through a costly screening procedure. This independence creates a channel through which students' preferences affect the strategic interaction between schools -- students with competing offers accept the most-preferred one, increasing the screening costs of unpopular schools. When preferences between schools are more heterogeneous, schools screen more, increasing the proportion of students with multiple offers, but paradoxically reducing the extent to which their preferences determine their outcomes. By observing the students' schools of origin, colleges can free-ride of the fierce competition that occurs during screening.