This paper established a novel multi-input multi-output (MIMO) communication network, in the presence of full-duplex (FD) transmitters and receivers with the assistance of dual-side intelligent omni surface. Compared with the traditional IOS, the dual-side IOS allows signals from both sides to reflect and refract simultaneously, which further exploits the potential of metasurfaces to avoid frequency dependence, and size, weight, and power (SWaP) limitations. By considering both the downlink and uplink transmissions, we aim to maximize the weighted sum rate, subject to the transmit power constraints of the transmitter and the users and the dual-side reflecting and refracting phase shifts constraints. However, the formulated sum rate maximization problem is not convex, hence we exploit the weighted minimum mean square error (WMMSE) approach, and tackle the original problem iteratively by solving two sub-problems. For the beamforming matrices optimizations of the downlink and uplink, we resort to the Lagrangian dual method combined with a bisection search to obtain the results. Furthermore, we resort to the quadratically constrained quadratic programming (QCQP) method to optimize the reflecting and refracting phase shifts of both sides of the IOS. In addition, we introduce the case without a dual-side IOS for comparison. Simulation results validate the efficacy of the proposed algorithm and demonstrate the superiority of the dual-side IOS.