Metallic nanoscale contacts realized, e.g., in scanning tunneling microscopy support localized surface-plasmon polaritons (LSPs) which emit light upon excitation by a tunnel current. We develop a theoretical framework based on the Keldysh nonequilibrium Green's function formalism to address this problem and demonstrate how the light emission is a direct probe of the finite-frequency current noise of the biased contact. From a perturbative calculation of the finite-frequency noise to higher order in the interaction between the tunnel current and the LSP, we find that experimentally observed above-threshold light emission--light with frequency higher than the applied bias--can be explained by a plasmon-induced electron-electron interaction which generates "hot electrons" in the contact.