Hierarchical multivariate stochastic differential equations (SDEs) are commonly used in many areas, such as ecology, finance, biological science and oceanography. Statistical inference based on discretely observed data requires estimating the transition density which is unknown for most SDE models. Typically, one would estimate the transition density and use the approximation for statistical inference. However, many estimation methods will fail when the observations are too sparse or when the models have a hierarchical structure. In this work, we use a Bayesian approach to explore the posterior distribution of the SDE model parameters. In the MCMC algorithm we use data augmentation to understand how the approximation of the transition density affects the inference procedure. We give guidelines on balancing the computational demands with the need to provide reliable and accurate posterior inference. Simulations are used to evaluate these guidelines. We demonstrate these methods on the analysis of animal tracking data.