Dependent data underlies many statistical studies in the social and health sciences, which often involve sensitive or private information. Differential privacy (DP) and in particular \textit{user-level} DP provide a natural formalization of privacy requirements for processing dependent data where each individual provides multiple observations to the dataset. However, dependence introduced, e.g., through repeated measurements challenges the existing statistical theory under DP-constraints. In \iid{} settings, noisy Winsorized mean estimators have been shown to be minimax optimal for standard (\textit{item-level}) and \textit{user-level} DP estimation of a mean $μ\in \R^d$. Yet, their behavior on potentially dependent observations has not previously been studied. We fill this gap and show that Winsorized mean estimators can also be used under dependence for bounded and unbounded data, and can lead to asymptotic and finite sample guarantees that resemble their \iid{} counterparts under a weak notion of dependence. For this, we formalize dependence via log-Sobolev inequalities on the joint distribution of observations. This enables us to adapt the stable histogram by Karwa and Vadhan (2018) to a non-\iid{} setting, which we then use to estimate the private projection intervals of the Winsorized estimator. The resulting guarantees for our item-level mean estimator extend to \textit{user-level} mean estimation and transfer to the local model via a randomized response histogram. Using the mean estimators as building blocks, we provide extensions to random effects models, longitudinal linear regression and nonparametric regression. Therefore, our work constitutes a first step towards a systematic study of DP for dependent data.