We present, for the first time, scalable polarizable molecular dynamics (MD) simulations within a polarizable continuum solvent with molecular shape cavities and exact solution of the mutual polarization. The key ingredients are a very efficient algorithm for solving the equations associated with the polarizable continuum, in particular, the domain decomposition Conductor-like Screening Model (ddCOSMO), which involves a rigorous coupling of the continuum with the polarizable force field achieved through a robust variational formulation and an effective strategy to solve the coupled equations. The coupling of ddCOSMO with nonvariational force fields, including AMOEBA, is also addressed. The MD simulations are feasible, for real-life systems, on standard cluster nodes; a scalable parallel implementation allows for further acceleration in the context of a newly developed module in Tinker, named Tinker-HP. NVE simulations are stable, and long-term energy conservation can be achieved. This paper is focused on the methodological developments, the analysis of the algorithm, and the stability of the simulations; a proof-of-concept application is also presented to attest to the possibilities of this newly developed technique.