As network data has become increasingly prevalent, a substantial amount of attention has been paid to the privacy issue in publishing network data. One of the critical challenges for data publishers is to preserve the topological structures of the original network while protecting sensitive information. In this paper, we propose a personalized edge flipping mechanism that allows data publishers to protect edge information based on each node's privacy preference. It can achieve differential privacy while preserving the community structure under the multi-layer degree-corrected stochastic block model after appropriately debiasing, and thus consistent community detection in the privatized multi-layer networks is achievable. Theoretically, we establish the consistency of community detection in the privatized multi-layer network and show that better privacy protection of edges can be obtained for a proportion of nodes while allowing other nodes to give up their privacy. Furthermore, the advantage of the proposed personalized edge-flipping mechanism is also supported by its numerical performance on various synthetic networks and a real-life multi-layer network.