Power-to-gas (PtG), a technology that converts electricity into hydrogen, is expected to become a core component of future low-carbon energy systems. While its economics and performance as a sector coupling technique have been well studied in the context of perfectlycompetitive energy markets, the distortions caused by the presence of large strategic players with a multi-market presence have received little attention. In this paper, we examine them by specifying a partial equilibrium model that provides a stylized representation of the interactions among the natural gas, electricity, and hydrogen markets. Using that model, we compare several possible ownership organizations for PtG to investigate how imperfect competition affects its operations. Evidence gained from these market simulations show that the effects of PtG vary with the multi-market profile of its operator. Producers of fossil-based hydrogen tend to make little use of PtG, whereas renewable power producers use it more to increase the electricity prices. Although PtG operations are profitable and can be welfare-enhancing, the social gain is either very tiny or negative when PtG is strategically operated in conjunction with variable renewable generation. In that case, PtG also raises environmental concerns as it stimulates the use of polluting thermoelectric generation.