Coded caching is a technique where we utilize multi-casting opportunities to reduce rate in cached networks. One limitation of coded caching schemes is that they reveal the demands of all users to their peers. We consider coded caching schemes that assure privacy for user demands with a particular focus on reducing subpacketization. For the 2-user, 2-file case, we present a new linear demand-private scheme with the lowest possible subpacketization. This is done by presenting the scheme explicitly and proving impossibility results under lower subpacketization. We also generalize demand privacy to various settings. First, we extend demand private schemes to two-tier hierarchical caching networks. Second, we propose a new scheme that combines demand privacy and content security where user demands and file contents are hidden from unintended parties. Third, we propose coded caching schemes with non-uniform privacy which can significantly reduce subpacketization by trading it off with privacy.