Context. Rapidly damped transverse oscillations of coronal loop systems are often observed. Aims. We aim to study analytically the resonantly damped oscillations of a system of two not necessarily identical coronal loops and their dependence on the equilibrium parameters, improving on and extending the results for two identical coronal loops. Methods. The linearised magnetohydrodynamic equations for a cold plasma were solved in the long-wavelength limit and for thin boundary layers in bicylindrical coordinates. We investigated the effects of the density contrast between the two loops, the thickness of their inhomogeneous layers, and the separation distance between them. The complex spectrum was also studied. Results. We obtained more general expressions for the linear damping rate of the transverse oscillations in a system of two coronal loops. The results can be reduced to expressions found previously for the special cases of one vanishing loop or two identical loops. The interaction between the loops results in a stronger damping of the high-frequency eigenoscillation in comparison with that of the low-frequency eigenoscillation. By decreasing the distance between loops, the efficiency of resonant damping is reduced.