Twin vertices in simple unweighted graphs are vertices that have the same neighbours and, in the case of weighted graphs with possible loops, the corresponding incident edges have equal weights. In this paper, we explore the role of twin vertices in quantum state transfer. In particular, we provide characterizations of periodicity, perfect state transfer, and pretty good state transfer between twin vertices in a weighted graph with respect to its adjacency, Laplacian and signless Laplacian matrices. As an application, we provide characterizations of all simple unweighted double cones on regular graphs that exhibit periodicity, perfect state transfer, and pretty good state transfer.