Learning driving behaviour from observed trajectories of real-world road users is subject to observation noise. A lack of proper treatment of such noise confounds the learned behaviour, and can lead to unrealistic simulation of agents for testing Autonomous Vehicles (AVs). We use a state-space model as the underlying data generating process of our dataset of observed trajectories, and jointly learn its transition and observation models using a particle approximation of the score function. Our method yields a consistent estimate of the score without having to differentiate through the particle filter. We also demonstrate the efficacy of our learned transition and observation models as generative models for sampling driving behaviour and observation noise in simulation respectively.