The Wilcoxon rank-sum test is one of the most popular distribution-free procedures for testing the equality of two univariate probability distributions. One of the main reasons for its popularity can be attributed to the remarkable result of Hodges and Lehmann (1956), which shows that the asymptotic relative efficiency of Wilcoxon's test with respect to Student's $t$-test, under location alternatives, never falls below 0.864, despite the former being exactly distribution-free for all sample sizes. Even more striking is the result of Chernoff and Savage (1958), which shows that the efficiency of a Gaussian score transformed Wilcoxon's test, against the $t$-test, is lower bounded by 1. In this paper we study the two-sample problem in the multivariate setting and propose distribution-free analogues of the Hotelling $T^2$ test (the natural multidimensional counterpart of Student's $t$-test) based on optimal transport and obtain extensions of the above celebrated results over various natural families of multivariate distributions. Our proposed tests are consistent against a general class of alternatives and satisfy Hodges-Lehmann and Chernoff-Savage-type efficiency lower bounds, despite being entirely agnostic to the underlying data generating mechanism. In particular, a collection of our proposed tests suffer from no loss in asymptotic efficiency, when compared to Hotelling $T^2$. To the best of our knowledge, these are the first collection of multivariate, nonparametric, exactly distribution-free tests that provably achieve such attractive efficiency lower bounds. We also demonstrate the broader scope of our methods in optimal transport based nonparametric inference by constructing exactly distribution-free multivariate tests for mutual independence, which suffer from no loss in asymptotic efficiency against the classical Wilks' likelihood ratio test, under Konijn alternatives.