We investigate the ground-state properties, the far-infrared (FIR) response, and the collective modes of a square lattice of double-layered quantum dots by applying classical and quantum-mechanical concepts. Using classical self-consistent linear response theory for the dot array we derive analytic results for the magnetoabsorption spectrum and the frequencies, linewidths, and oscillator strengths of the optical and acoustic collective modes including the effect of intercell and intracell interactions. Within the full quantum-mechanical calculation (applying density-functional theory with the local-density approximation) for a single double-layered quantum dot we obtain numerically ground-state properties and the FIR excitation frequencies. We use a quantum dot model with a realistic distribution of background charge, which accounts for surface states and total charge neutrality. In both approaches we study the dependence of the oscillator strengths of the acoustic mode on the asymmetry of the double-layered system in order to give information on the condition for experimental detection.