The first operation of a coaxial free-electron maser (FEM) based on two-dimensional (2D) distributed feedback has been recently observed. Analytical and numerical modeling, as well as measurements, of microwave radiation generated by a FEM with a cavity defined by coaxial structures with a 2D periodic perturbation on the inner surfaces of the outer conductor were carried out. The two-mirror cavity was formed with two 2D periodic structures separated by a central smooth section of coaxial waveguide. The FEM was driven by a large diameter (7 cm), high-current (500 A), annular electron beam with electron energy of 475 keV. Studies of the FEM operation have been conducted. It has been demonstrated that by tuning the amplitude of the undulator or guide magnetic field, modes associated with the different band gaps of the 2D structures were excited. The Ka-band FEM generated 15 MW of radiation with a 6% conversion efficiency, in good agreement with theory.