Let $(X(t), t\ge 0)$ be defined by a stochastic differential equation including a random effect $\phi$ in the drift and diffusion coefficients. We characterize the stationary distributions of the joint process $((\phi, X(t)), t\ge 0)$ which are non unique and prove limit theorems and central limit theorems for functionals of the sample path $(X(t), t\in [ 0, T])$ as $T$ tends to infinity. This allows to build several estimators of the random variable $\phi$ which are consistent and asymptotically mixed normal with rate $\sqrt{T}$. Examples are given fulfilling the assumptions of the limit theorems. Parametric estimation of the distribution of the random effect from $N$ {\em i.i.d.} processes $(X_j(t), t\in [ 0, T]), j=1,\ldots,N$ is considered. Parametric estimators are built and proved to be $\sqrt{N}$-consistent and asymptotically Gaussian as both $N$ and $T=T(N)$ tend to infinity with $T(N)/N$ tending to infinity.