We have studied electron transport in two closely spaced, but electrically isolated, two-dimensional electron gases (2DEGs) formed in GaAs/AlGaAs double quantum wells. A frictional drag voltage is found to be induced in one layer when a current is passed through the other. This voltage represents a direct measure of the interlayer electron-electron interactions, since it is determined by the momentum transfer due to scattering events between electrons in the different layers. We present new experimental data, based on a detailed carrier density dependence of these electron-electron interactions. Our results strongly support a recent theory proposing an interaction via virtual acoustic phonons. We have shown, in particular, that the 'phonon part' of the interlayer scattering rate is a function of the relative electron densities in the layers, with a maximum scattering rate when the densities are matched.