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The coupled electricity and heat system (CEHS) is considered one of the most efficient energy utilization schemes for the energy transition to solve energy crises and environmental problems. However, the individual data between the power system (PS) and heat system (HS) probably limit the high share and optimization operation. Moreover, the uncertain renewables and complex coupled network create challenges regarding the operation risk of CEHS. Given that the CEHS may be affiliated with different operation entities, this paper proposes two operation modes to achieve the solution of the optimal risk operation model (OROM), including the distributionally robust chance constraint (DRCC)-based centralized risk operation mode (CROM) and the DRCC-based alternating direction method of multipliers (ADMM) distributed risk operation mode (DROM). By formulating the moment-based ambiguity set estimated from historical data and introducing the operation risk constraints, a tractable reformulation of two-stage DRCC OROM problems is presented under CROM. Moreover, the DRCC-based ADMM distributed algorithm with the guarantee of convergence is developed to optimize the PS and HS independently under DROM considering the operation risk. The two operation modes achieve the minimum amount of information shared between the two systems regarding the underlying distribution. In the numerical case, the approximate OROM solution is obtained between CROM and DROM. The two proposed approaches based on different operation modes outperform the existing methods according to the risk level depicted by different forms of the violation probability and the risk cost compared with Gaussian chance constraint (GCC) under CROM, while the safety coefficient $\epsilon $ is set as 0.25. Then, the impact of the iteration number on the convergence is also discussed with comparison of the classical ADMM under DROM.