We study the problem of estimating non-linear functionals of discrete distributions in the context of local differential privacy. The initial data $x_1,\ldots,x_n \in[K]$ are supposed i.i.d. and distributed according to an unknown discrete distribution $p = (p_1,\ldots,p_K)$. Only $\alpha$-locally differentially private (LDP) samples $z_1,...,z_n$ are publicly available, where the term 'local' means that each $z_i$ is produced using one individual attribute $x_i$. We exhibit privacy mechanisms (PM) that are interactive (i.e. they are allowed to use already published confidential data) or non-interactive. We describe the behavior of the quadratic risk for estimating the power sum functional $F_{\gamma} = \sum_{k=1}^K p_k^{\gamma}$, $\gamma >0$ as a function of $K, \, n$ and $\alpha$. In the non-interactive case, we study twol plug-in type estimators of $F_{\gamma}$, for all $\gamma >0$, that are similar to the MLE analyzed by Jiao et al. (2017) in the multinomial model. However, due to the privacy constraint the rates we attain are slower and similar to those obtained in the Gaussian model by Collier et al. (2020). In the sequentially interactive case, we introduce for all $\gamma >1$ a two-step procedure which attains the parametric rate $(n \alpha^2)^{-1/2}$ when $\gamma \geq 2$. We give lower bounds results over all $\alpha-$LDP mechanisms and over all estimators using the private samples.

We find separation rates for testing multinomial or more general discrete distributions under the constraint of alpha-local differential privacy. We construct efficient randomized algorithms and test procedures, in both the case where only non-interactive privacy mechanisms are allowed and also in the case where all sequentially interactive privacy mechanisms are allowed. The separation rates are faster in the latter case. We prove general information theoretical bounds that allow us to establish the optimality of our algorithms among all pairs of privacy mechanisms and test procedures, in most usual cases. Considered examples include testing uniform, polynomially and exponentially decreasing distributions.