Differences in skill utilization across firms and labor markets have been associated with wage inequality, but whether this relationship reflects differences in worker or firm heterogeneity is still unclear. Combining linked employer-employee data from Italy with detailed information on skill demand extracted from online job vacancies, we study the relationship between wages and the demand for cognitive and social skills across labor markets defined by province, sector, and occupation. We then estimate the worker- and firm-pay components of the wage process through an AKM model and investigate their relationship with skill demand at the labor market level. We find a strong and positive association between wages and the demand for cognitive and social skills, which is pronounced when both skills are required jointly for the same job position revealing their complementarity. Our decomposition suggests that higher wages in markets in which firms more frequently demand cognitive and social skills jointly are driven by worker effects, reflecting the higher market value of a hybrid skill set, rather than by firm pay policies. Conversely, markets in which firms demand more of either cognitive or social skills only are associated to higher firm effects, suggesting that more rents are shared to specialized workers, while the market value of these specialized skills for workers is lower. These results highlight the role of worker and firm heterogeneity as channels through which skill demand differences affect overall wage inequality.