This paper studies the relationship between wage inequality and skill demand and its connection to worker and firm heterogeneity. Combining linked employer-employee data from Italy with detailed information on skill demand extracted from online job vacancies, we first study the relationship between wages and the demand for cognitive and social skills across labor markets defined by province, sector, and occupation. 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 highlighting their complementarity. 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. Our decomposition suggests that in markets in which firms demand more frequently both cognitive and social skills, higher wages are predominantly attributed to worker effects, reflecting the higher market value of a combined skill set, rather than by firm pay policies. In contrast, in markets where firms predominantly demand either cognitive or social skills, higher wages are associated with higher firm effects, indicating more rents for specialized workers, despite these specialized skills having a lower market value for workers. These results highlight the role of worker and firm heterogeneity as mechanisms through which variations in skill demand influence overall wage inequality.