Peculiarities of broadband ferromagnetic resonance response of conducting magnetic bilayer films have been studied theoretically. The film thickness was assumed to be smaller than the microwave magnetic skin depth. Excitation of magnetization precession by a homogeneous microwave field incident from one surface of the film has been considered, which mimics the microwave field of a wide microstrip transducer. Strong asymmetry of the response has been found. Depending on the order of layers with respect to the transducer location either the first higher-order standing spin-wave mode or the fundamental mode shows the largest response. This effect is explained as shielding of the transducer field by eddy currents in the films. This shielding remains very efficient for films with thicknesses below the microwave magnetic skin depth. This effect may be useful for studying buried magnetic interfaces and should be accounted for in future development of broadband inductive ferromagnetic resonance methods.