Cell-free systems significantly improve network capacity by enabling joint user service without cell boundaries, eliminating inter-cell interference. However, to satisfy further capacity demands, it leads to high-cost problems of both hardware and power consumption. In this paper, we investigate multiple reconfigurable intelligent surfaces (RISs) aided cell-free systems where RISs are introduced to improve spectrum efficiency in an energy-efficient way. To overcome the centralized high complexity and avoid frequent information exchanges, a cooperative distributed beamforming design is proposed to maximize the weighted sum-rate performance. In particular, the alternating optimization method is utilized with the distributed closed-form solution of active beamforming being derived locally at access points, and phase shifts are obtained centrally based on the Riemannian conjugate gradient (RCG) manifold method. Simulation results verify the effectiveness of the proposed design whose performance is comparable to the centralized scheme and show great superiority of the RISs-aided system over the conventional cellular and cell-free system.