Energy flow calculation (EFC) plays an important role in steady-state analysis of multi-energy systems (MESs). However, the independent management of sub-energy systems (subsystems) poses a considerable challenge to solve the high-order nonlinear energy flow model due to the limited information exchange between these subsystems. In this article, a fixed-point based distributed method is proposed for EFC in an electricity-gas-heating system. Firstly, the mathematical modeling of each subsystem with coupling units is introduced. Then, two information exchange structures among subsystems are presented as sequential and parallel structures. Based on the fixed-point theorem, novel distributed sequential and parallel methods for EFC are proposed to calculate energy flow distribution in MESs. In our proposed method, the EFC in subsystems is implemented by the individual system operators, with limited information exchange between subsystems. Therefore, the information privacy of subsystems can be preserved in this solution process. Moreover, the convergence of the proposed method is guaranteed, and the sufficient conditions for the convergence are presented. Lastly, simulations on a MES demonstrate the effectiveness of the proposed method and the quantified superiority over the existing methods in computation time, accuracy and reliability.