The paper proposes a cooperative scheme for the coexistence of a multiple-input-multiple-output (MIMO) communication system and a matrix completion (MC) based, collocated MIMO (MIMO-MC) radar. To facilitate the coexistence, and also deal with clutter, both the radar and the communication systems use transmit precoding. For waveform flexibility, the radar uses a random unitary waveform matrix. We prove that for such waveforms and any precoding matrix, the error performance of MC is guaranteed. The radar transmit precoder, the radar subsampling scheme, and the communication transmit covariance matrix are jointly designed in order to maximize the radar SINR, while meeting certain communication rate and power constraints. The joint design is implemented at a control center, which is a node with whom both systems share physical layer information, and which also performs data fusion for the radar. We provide efficient algorithms for the proposed optimization problem, along with insight on the feasibility and properties of the proposed design. Simulation results show that the proposed scheme significantly improves the spectrum sharing performance in various scenarios.