The student-optimal stable mechanism (DA), the most popular mechanism in school choice, is the only one that is stable and strategy-proof. However, when DA is implemented, a student can change the schools of others without changing her own. We show that this drawback is limited: a student cannot change her schoolmates while remaining at the same school. We refer to this new property as local non-bossiness and use it to provide a new characterization of DA that does not rely on stability. Furthermore, we show that local non-bossiness plays a crucial role in providing incentives to be truthful when students have preferences over their colleagues. As long as students have school-lexicographic preferences over colleagues—meaning that they first consider the school to which they are assigned and then their schoolmates—DA induces the only stable and strategy-proof mechanism. There is limited room to expand this preference domain without compromising the existence of a stable and strategy-proof mechanism.