This talk studies the value of randomized solutions (VRS) in distributionally robust mixed integer programming problems. We show different methods for obtaining upper bounds on VRS and identify conditions under which some of them are tight. We also devise and implement a column-generation algorithm for identifying optimal randomized solutions in two-stage distributionally robust optimization with right-hand-side uncertainty. We empirically illustrate our findings in a capacitated facility location problem where the distribution is known to be part of a Wasserstein ambiguity set. This is joint work with Ahmed Saif.