A twin dominating set of a digraph D is a subset S of vertices if, for every vertex u ∉ S, there are vertices x,y ∈ S such that ux and yu are arcs of D. A digraph D is round if the vertices can be labeled as v₀,v₁,...,v_n-1 so that, for each vertex v_i, the out-neighbors of v_i appear consecutively following v_i and the in-neighbors of v_i appear consecutively preceding v_i. In this paper, we give polynomial time algorithms for finding a minimum weight twin dominating set and a minimum weight total twin dominating set for a weighted round digraph. Then we show that there is a polynomial time algorithm for deciding whether a locally semicomplete digraph has an independent twin dominating set. The class of locally semicomplete digraphs contains round digraphs as a special case.