The quantum dynamics of open many-body systems poses a challenge for computational approaches. Here we develop a stochastic scheme based on the positive P phase-space representation to study the nonequilibrium dynamics of coupled spin-boson networks that are driven and dissipative. Such problems are at the forefront of experimental research in cavity and solid state realizations of quantum optics, as well as cold atom physics, trapped ions and superconducting circuits. We demonstrate and test our method on a driven, dissipative two-site system, each site involving a spin coupled to a photonic mode, with photons hopping between the sites, where we find good agreement with Monte Carlo Wavefunction simulations. In addition to numerically reproducing features recently observed in an experiment [Phys. Rev. X 4, 031043 (2014)], we also predict a novel steady state quantum dynamical phase transition for an asymmetric configuration of drive and dissipation.