In this article we investigate consistency and asymptotic normality of the maximum likelihood and the posterior distribution of the parameters in the context of state space stochastic differential equations (SDEs). We then extend our asymptotic theory to random effects models based on systems of state space SDEs, covering both independent and identical and independent but non-identical collections of state space SDEs. We also address asymptotic inference in the case of multidimensional linear random effects, and in situations where the data are available in discretized forms. It is important to note that asymptotic inference, either in the classical or in the Bayesian paradigm, has not been hitherto investigated in state space SDEs.