We studied a sample of 274 radio and X-ray selected quasars (XQSOs) detected in the COSMOS and XXL-S radio surveys at 3 GHz and 2.1 GHz, respectively. This sample was identified by adopting a conservative threshold in X-ray luminosity, LX [2-10 keV] ≥ 1044 erg s-1, selecting only the most powerful quasars. A number of previous studies on the origin of radio emission in type-1 quasars have focused on the radio loudness distributions, some claiming to have found evidence for bimodality, pointing toward the existence of two physically different mechanisms for the radio emission. Using available multiwavelength data, we examined various criteria for the selection of radio-loud (RL) and radio-quiet (RQ) XQSOs and found that the number of RL/RQ XQSOs changes significantly depending on the chosen criterion. This discrepancy arises due to the different criteria tracing different physical processes and due to the fact that our sample was selected from flux-limited radio and X-ray surveys. Another approach to study the origin of radio emission in XQSOs is via their radio luminosity functions (RLF). We constructed the XQSO 1.4 GHz RLFs in six redshift bins at 0.5 ≤ z ≤ 3.75. The lower-1.4 GHz luminosity end shows a higher normalization than expected only from AGN contribution in all studied redshift bins. We found that the so-called “bump” is mostly dominated by emission due to star-forming processes within the host galaxies of XQSOs. As expected, AGN-related radio emission is the dominant contribution at the higher-luminosity end of RLF. To study the evolution of the XQSO RLF, we used a combination of analytic forms from the literature to constrain the “bump” due to star formation and the higher-luminosity AGN part of the RLF. We defined two 1.4 GHz luminosity thresholds, Lth, SF and Lth, AGN, below and above which more than 80% of sources contributing to the RLF are dominated by star formation and AGN-related activity, respectively. The two thresholds evolve with redshift, which is most likely driven by the strong evolution of star formation rates of the XQSO host galaxies. We found that both the lower and higher luminosity ends evolve significantly in density, while their luminosity evolution parameters are consistent with being constant. We found that the lower-luminosity end evolves both in density and luminosity, while the higher-luminosity end evolves significantly only in density. Our results expose the dichotomy of the origin of radio emission: while the higher-luminosity end of the XQSO RLF is dominated by AGN activity, the lower-luminosity end is dominated by the star formation-related processes.