This paper focuses on jointly inferring network and community structures from the dynamics of complex systems. Although many approaches have been designed to solve these two problems solely, none of them consider explicit shareable knowledge across these two tasks. Community detection (CD) from dynamics and network reconstruction (NR) from dynamics are natural synergistic tasks that motivate the proposed evolutionary multitasking NR and CD framework, called network collaborator (NC). In the process of NC, the NR task explicitly transfers several better network structures for the CD task, and the CD task explicitly transfers a better community structure to assist the NR task. Moreover, to transfer knowledge from the NR task to the CD task, NC models the study of CD from dynamics to find communities in the dynamic network and then considers whether to transfer knowledge across tasks. A test suite for multitasking NR and CD problems (MTNRCDPs) is designed to verify the performance of NC. The experimental results conducted on the designed MTNRCDPs have demonstrated that joint NR with CD has a synergistic effect, where the network structure used to inform the existence of communities is also inherently employed to improve the reconstruction accuracy, which, in turn, can better demonstrate the discovering of the community structure. The code is available at: https://github.com/xiaofangxd/EMTNRCD.