We consider a single-cell scenario involving a single base station (BS) with a massive array serving multi-antenna terminals in the downlink of a mmWave channel. We present a class of multiuser user MIMO schemes, which rely on uplink training from the user terminals, and on uplink/downlink channel reciprocity. The BS employs virtual sector-based processing according to which, user-channel estimation and data transmission are performed in parallel over non-overlapping angular sectors. The uplink training schemes we consider are non-orthogonal, that is, we allow multiple users to transmit pilots on the same pilot dimension (thereby potentially interfering with one another). Elementary processing allows each sector to determine the subset of user channels that can be resolved on the sector (effectively pilot contamination free) and, thus, the subset of users that can be served by the sector. This allows resolving multiple users on the same pilot dimension at different sectors, thereby increasing the overall multiplexing gains of the system. Our analysis and simulations reveal that, by using appropriately designed directional training beams at the user terminals, the sector-based transmission schemes we present can yield substantial spatial multiplexing and ergodic user-rates improvements with respect to their orthogonal-training counterparts.