  The interaction between molecules is commonly ignored in surface-enhanced
Raman scattering (SERS). Under this assumption, the total SERS signal is
described as the sum of the individual contributions of each molecule treated
independently. We adopt here an optomechanical description of SERS within a
cavity quantum electrodynamics framework to study how collective effects emerge
from the quantum correlations of distinct molecules. We derive analytical
expressions for identical molecules and implement numerical simulations to
analyze two types of collective phenomena: (i) a decrease of the laser
intensity threshold to observe strong non-linearities as the number of
molecules increases, within intense illumination, and (ii) identification of
superradiance in the SERS signal, namely a quadratic scaling with the number of
molecules. The laser intensity required to observe the latter in the
anti-Stokes scattering is relatively moderate, which makes it particularly
accessible to experiments. Our results also show that collective phenomena can
survive in the presence of moderate homogeneous and inhomogeneous broadening.
