We study phonon-mediated frictional drag between two two-dimensional electron gases. We find two different regimes depending on whether the phonon mean free path ℓ_ph is larger or smaller than a characteristic value ℓ_{ph, crit}. For ℓ_ph > ℓ_{ph, crit}, the drag is dominated by coupling through an electron-phonon collective mode, whereas for ℓ_ph < ℓ_{ph, crit}, the drag is mediated by damped phonons. In the latter case, the drag varies logarithmically with interlayer distance. ℓ_{ph, crit} depends on the electron densities, and it is therefore possible to experimentally tune a given system between the two regimes.