Recent years have seen unprecedented advance in the design and control of quantum computers. Nonetheless, their applicability is still restricted and access remains expensive. Therefore, a substantial amount of quantum algorithms research still relies on simulating quantum circuits on classical hardware. However, due to the sheer complexity of simulating real quantum computers, many simulators unrealistically simplify the problem and instead simulate perfect quantum hardware, i.e., they do not consider errors caused by the fragile nature of quantum systems. Stochastic quantum simulation provides a conceptually suitable solution to this problem: physically motivated errors are applied in a probabilistic fashion throughout the simulation. In this work, we propose to use decision diagrams, as well as concurrent executions, to substantially reduce resource-requirements-which are still daunting - for stochastic quantum circuit simulation. Backed up by rigorous theory, empirical studies show that this approach allows for a substantially faster and much more scalable simulation for certain quantum circuits.