The paper investigates a new framework for spectrum sharing between a MIMO-MC radar (MIMO radar using matrix completion) and a MIMO communication system, based on radar transmit precoding. The radar transmit precoder is jointly designed with the communication codewords so that the SINR at the radar receiver is maximized while meeting certain rate and power constraints at the communication system. By shaping the transmit beam, the proposed approach results in enhanced SINR at the receive antennas. Unlike prior works, there is no need for sharing the transmit waveforms with the communication system; only the precoding matrix needs to be shared. Therefore, the proposed scheme is less vulnerable to adversaries. Simulation results demonstrate that the proposed method improves the radar SINR and the matrix completion accuracy over previous approaches.