In this paper, we consider partially observed dynamical systems where the observations are given as the sum of latent variables lying in a general state space and some independent noise with unknown distribution. In the case of dependent latent variables such as Markov chains, it is shown that this fully nonparametric model is identifiable with respect to both the distribution of the latent variables and the distribution of the noise, under mostly a light tail assumption on the latent variables. Two nonparametric estimation methods are proposed and we prove that the corresponding estimators are consistent for the weak convergence topology. These results are illustrated with numerical experiments.