Although the benefits of precoding and combining of data signals are widely recognized, the potential of these techniques for pilot transmission is not fully understood. This is particularly relevant for multiuser multiple-input multiple-output (MU-MIMO) cellular systems using millimeter-wave (mmWave) communications, where multiple antennas will have to be used both at the transmitter and the receiver to overcome the severe path loss. In this paper, we characterize the gains of pilot precoding and combining in terms of channel estimation quality and achievable data rate. Specifically, we consider three uplink pilot transmission scenarios in a mmWave MU-MIMO cellular system: 1) non-precoded and uncombined, 2) precoded but uncombined, and 3) precoded and combined. We show that a simple precoder that utilizes only the second-order statistics of the channel reduces the variance of the channel estimation error by a factor that is proportional to the number of user equipment (UE) antennas. We also show that using a linear combiner designed based on the second-order statistics of the channel significantly reduces multiuser interference and provides the possibility of reusing some pilots. Specifically, in the large antenna regime, pilot precoding and combining help to accommodate a large number of UEs in one cell, significantly improve channel estimation quality, boost the signal-to-noise ratio of the UEs located close to the cell edges, alleviate pilot contamination, and address the imbalanced coverage of pilot and data signals.