This paper studies the behaviour of the stochastic subgradient descent (SSGD) method applied to over-parameterized nonsmooth optimization problems that satisfy an interpolation condition. By leveraging the composite structure of the empirical risk minimization problems, we prove that when interpolation holds, SSGD converges at the same rate as the stochastic gradient descent (SGD) method applied to smooth problems. Our analysis provides a partial explanation for the empirical observation that sometimes SGD and SSGD behave similarly for training smooth and nonsmooth machine learning models. We also prove that the rates we derive are optimal for the subgradient method in the convex and interpolation setting.