Gini's rank association coefficient and Spearman's footrule, as statistics for testing independence in bivariate samples, are as natural as Spearman's and Kendall's rank correlation coefficients, but their efficiency properties are not well explored. We find here the expression for the local Bahadur efficiency of Gini's test and Spearman's footrule for general alternatives. Several examples are given in which both statistics behave better than Spearman's and Kendall's coefficients. Similar results are obtained the general measure of monotone dependence considered recently by Conti et al. (1996). The coincidence between Pitman and Bahadur efficiencies is also proved.