In this paper, we analyze the problem of optimally allocating resources in a distributed and privacy-preserving manner. We propose a novel distributed optimal resource allocation algorithm with privacy-preserving guarantees, which operates over a directed communication network. Our algorithm converges in finite time and allows each node to process and transmit quantized messages. Our algorithm utilizes a distributed quantized average consensus strategy combined with a privacy-preserving mechanism. We show that the algorithm converges in finite-time, and we prove that, under specific conditions on the network topology, nodes are able to preserve the privacy of their initial state. Finally, to illustrate the results, we consider an example where test kits need to be optimally allocated proportionally to the number of infections in a region. It is shown that the proposed privacy-preserving resource allocation algorithm performs well with an appropriate convergence rate under privacy guarantees.