Given a directed graph $G$ and integers $k$ and $l$, a D-core is the maximal subgraph $H \subseteq G$ such that for every vertex of $H$, its in-degree and out-degree are no smaller than $k$ and $l$, respectively. For a directed graph $G$, the problem of D-core decomposition aims to compute the non-empty D-cores for all possible values of $k$ and $l$. In the literature, several \emph{peeling-based} algorithms have been proposed to handle D-core decomposition. However, the peeling-based algorithms that work in a sequential fashion and require global graph information during processing are mainly designed for \emph{centralized} settings, which cannot handle large-scale graphs efficiently in distributed settings. Motivated by this, we study the \emph{distributed} D-core decomposition problem in this paper. We start by defining a concept called \emph{anchored coreness}, based on which we propose a new H-index-based algorithm for distributed D-core decomposition. Furthermore, we devise a novel concept, namely \emph{skyline coreness}, and show that the D-core decomposition problem is equivalent to the computation of skyline corenesses for all vertices. We design an efficient D-index to compute the skyline corenesses distributedly. We implement the proposed algorithms under both vertex-centric and block-centric distributed graph processing frameworks. Moreover, we theoretically analyze the algorithm and message complexities. Extensive experiments on large real-world graphs with billions of edges demonstrate the efficiency of the proposed algorithms in terms of both the running time and communication overhead.