We analyse the cluster luminosity functions (CLFs) of the youngest star clusters in two galaxies exhibiting prominent circumnuclear starburst rings. We focus specifically on NGC 1512 and NGC 6951, for which we have access to H{alpha} data that allow us to unambiguously identify the youngest sample clusters. To place our results on a firm statistical footing, we first explore in detail a number of important technical issues affecting the process from converting the observational data into the spectral energy distributions of the objects in our final catalogues. The CLFs of the young clusters in both galaxies exhibit approximate power-law behaviour down to the 90 per cent observational completeness limits, thus showing that star cluster formation in the violent environments of starburst rings appears to proceed similarly as that elsewhere in the local Universe. We discuss this result in the context of the density of the interstellar medium in our starburst-ring galaxies.