Millimeter wave (mmWave) systems will likely employ directional beamforming with large antenna arrays at both the transmitters and receivers. Acquiring channel knowledge to design these beamformers, however, is challenging due to the large antenna arrays and small signal-to-noise ratio before beamforming. In this paper, we propose and evaluate a downlink system operation for multi-user mmWave systems based on compressed sensing channel estimation and conjugate analog beamforming. Adopting the achievable sum-rate as a performance metric, we show how many compressed sensing measurements are needed to approach the perfect channel knowledge performance. The results illustrate that the proposed algorithm requires an order of magnitude less training overhead compared with traditional lower-frequency solutions, while employing mmWave-suitable hardware. They also show that the number of measurements need to be optimized to handle the trade-off between the channel estimate quality and the training overhead.