Networked microgrid (NMG) exhibits noteworthy resiliency and flexibility benefits for the mutual support from neighboring microgrids. With high penetration of distributed energy resources (DERs) and the associated controls, the transient stability analysis of NMGs is of critical significance. To address the issues of computation burdens and privacy in the centralized transient analysis, this paper devises an ordinary differential equation (ODE)-enabled distributed transient stability (DTS) methodology for NMGs. First, an ODE-based microgrid model is established to capture the dynamics in the droop control of DERs as well as network and load. Further, a distributed DTS is devised for the ODE representation of an NMG, allowing a privacy-preserving transient analysis of each microgrid while accurately reconstructing the frequency dynamics under droop controls in all DERs. Extensive tests are performed to verify the validity of the ODE-based microgrid model through both dynamic response and eigenvalue analysis, and the efficacy of the DTS algorithm in simulating the large signal responses and the frequent fluctuations in NMG.