Noisy discretely observed diffusion processes with random drift function parameters are considered. Maximum likelihood and Bayesian estimation methods are extended to this model, respectively the Stochastic Approximation EM and the Gibbs sampler algorithms. They are based on the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measurement instants. A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in these two algorithms. Their convergence is proved. Errors induced on the likelihood and the posterior distribution by the Euler-Maruyama approximation are bounded as a function of the step size of the approximation. Results of a pharmacokinetic mixed model simulation study illustrate the accuracy of the maximum likelihood estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach.