In this paper, we study Simultaneous Communication of Data and Control (SCDC) information signals in Full Duplex (FD) Multiple-Input Multiple-Output (MIMO) wireless systems. In particular, considering an FD MIMO base station serving multiple single-antenna FD users, a novel multi-user communication scheme for simultaneous DownLink (DL) beamformed data transmission and UpLink (UL) pilot-assisted channel estimation is presented. Capitalizing on a recent FD MIMO hardware architecture with reduced complexity self-interference analog cancellation, we jointly design the base stations transmit and receive beamforming matrices as well as the settings for the multiple analog taps and the digital SI canceller with the objective to maximize the DL sum rate. Our simulation results showcase that the proposed approach outperforms its conventional half duplex counterpart with 50% reduction in hardware complexity compared to the latest FD-based SCDC schemes.