Technologies for mmWave communication are at the forefront of investigations in both industry and academia, as the mmWave band offers the promise of orders of magnitude additional available bandwidths to what has already been allocated to cellular networks. The much larger number of antennas that can be supported in a small footprint at mmWave bands can be leveraged to harvest massive-MIMO type beamforming and spatial multiplexing gains. Similar to LTE systems, two prerequisites for harvesting these benefits are detecting users and acquiring user channel state information (CSI) in the training phase. However, due to the fact that mmWave channels encounter much harsher propagation and decorrelate much faster, the tasks of user detection and CSI acquisition are both imperative and much more challenging than in LTE bands. In this paper, we investigate the problem of fast user detection and CSI acquisition in the downlink of small cell mmWave networks. We assume TDD operation and channel-reciprocity based CSI acquisition. To achieve densification benefits we propose pilot designs and channel estimators that leverage a combination of aggressive pilot reuse with fast user detection at the base station and compressed sensing channel estimation. As our simulations show, the number of users that can be simultaneously served by the entire mmWave-band network with the proposed schemes increases substantially with respect to traditional compressed sensing based approaches with conventional pilot reuse.