Are giants in random digraphs `almost' local?

Recently, the first author showed that the giant in random undirected graphs is `almost' local. This means that, under a necessary and sufficient condition, the limiting proportion of vertices in the giant converges in probability to the survival probability of the local limit. We extend this result to the setting of random digraphs, where connectivity patterns are significantly more subtle. For this, we identify the precise version of local convergence for digraphs that is needed. We also determine bounds on the number of strongly connected components, and calculate its asymptotics explicitly for locally tree-like digraphs, as well as for other locally converging digraph sequences under the `almost-local' condition for the strong giant. The fact that the number of strongly connected components is {\em not} local once more exemplifies the delicate nature of strong connectivity in random digraphs.