In this article, we propose an economic storage sharing framework for prosumers and energy storage providers (ESPs) to promote renewable energy utilization cooperatively. The optimal shared capacities of ESPs and the energy sharing profiles of prosumers are first derived via minimizing social energy costs. Then the storage sharing profits of ESPs and the energy sharing payments of prosumers are successively determined by the asymmetric bargaining-based benefits sharing model. Specifically, the prosumer group bargains with the ESPs with the nominal required capacity and the shared capacities as their bargaining power to share the storage sharing benefits. Then prosumers bargain with each other to share the energy sharing benefits with their bargaining power quantified by a nonlinear energy sharing mapping method. Therefore, the benefits sharing model based on the contributions of prosumers and ESPs is fair enough for the participants. Numerical simulation tests verify the efficiency of the proposed framework.