Comparing graphs by means of optimal transport has recently gained significant attention, as the distances induced by optimal transport provide both a principled metric between graphs as well as an interpretable description of the associated changes between graphs in terms of a transport plan. As the lack of symmetry introduces challenges in the typically considered formulations, optimal transport distances for graphs have mostly been developed for undirected graphs. Here, we propose two distance measures to compare directed graphs based on variants of optimal transport: (i) an earth movers distance (Wasserstein) and (ii) a Gromov-Wasserstein (GW) distance. We evaluate these two distances and discuss their relative performance for both simulated graph data and real-world directed cell-cell communication graphs, inferred from single-cell RNA-seq data.