We investigate the effects of private electric vehicles (EVs) on the resilience of distribution systems after disruptions. We propose a framework of network-based multi-agent optimization problems with equilibrium constraints (N-MOPEC) to consider the decentralized decision making of stakeholders in transportation and energy systems. To solve the high-dimensional non-convex problem, we develop an efficient computational algorithm based on exact convex reformulation. Numerical studies are conducted to illustrate the effectiveness of our modeling and computational approach and to draw policy insights. The proposed modeling and computational strategies could provide a solid foundation for the future study of power system resilience with private EVs in coupled transportation and power networks.