In this paper, the correlation between nearby user equipment (UE) is exploited, and a deep learning-based channel state information (CSI) feedback and cooperative recovery framework, CoCsiNet, is developed to reduce feedback overhead. The CSI information can be divided into two parts: shared by nearby UE and owned by individual UE. The key idea of exploiting the correlation is to reduce the overhead used to feedback the shared information repeatedly. Unlike in the general autoencoder framework, an extra decoder and a combination network are added at the base station to recover the shared information from the feedback CSI of two nearby UEs and combine the shared and individual information, respectively, but no modification is performed at the UEs. For a UE with multiple antennas, a baseline neural network architecture with long short-term memory modules is introduced to extract the correlation of nearby antennas. Given that the CSI phase is not sparse, two magnitude-dependent phase feedback strategies that introduce statistical and instant CSI magnitude information to the phase feedback process are proposed. Simulation results on two different channel datasets show the effectiveness of the proposed CoCsiNet.