When a single excitation is shared between a large number of two-level systems, a strong enhancement of the spontaneous emission appears. This phenomenon is known as superradiance. This enhanced rate can be accompanied by a shift of the emission frequency, the cooperative Lamb shift, issued from the exchange of virtual photons between the emitters. In this work we present a semiconductor optoelectronic device allowing the observation of these two phenomena at room temperature. We demonstrate experimentally and theoretically that plasma oscillations in spatially separated quantum wells interact through real and virtual photon exchange. This gives rise to a superradiant mode displaying a large cooperative Lamb shift.