High efficiency and high frequency are key research topics for the development of high power microwave devices. There are, however, serious issues with mode competition in relativistic Cerenkov devices based on a single electron beam, which can lead to low efficiency in the beam-wave interaction process. In this paper, a high-efficiency relativistic Cerenkov oscillator based on a dual electron beam is proposed. The hollow slow wave structure is embedded in the inner conductor of the coaxial waveguide to obtain a double coaxial electromagnetic structure, which can suppress mode competition and improve the beam-wave interaction efficiency of the dual-band radiation. Moreover, each band of radiation is independent of the other, such that high frequency output radiation is generated. Using the particle-in-cell method, the physical processes involving the beam-wave interaction in the oscillator are studied with the aim of designing a high-efficiency double-electromagnetic structure. Simulation results indicated that the output power levels for the X-band and Ka-band were 1.50 GW and 710 MW, respectively, and the efficiencies were 34.7% and 30.7%, respectively, for a diode voltage of 550 kV and a guiding magnetic field of 0.85 T.