Two-sample testing assesses whether two populations differ by comparing their probability distributions, with the Kolmogorov–Smirnov test as a classic example. While numerous extensions address multivariate data, modern applications increasingly involve complex objects such as probability distributions themselves. This leads to the problem of testing the equality of laws of random probability measures. We propose a distance-based twosample test for distinguishing laws of random probability measures using optimal transport theory, and leverage tools from empirical process theory to establish nonparametric theoretical guarantees. Empirically, we benchmark our method against existing approaches on simulated datasets and apply it to a mortality dataset.