In this paper we analyze the private and public incentives towards skill acquisition when the skill level of workers determines the quality level of goods, and both labor and product markets are non competitive. We delve into the mechanisms that determine the equilibrium skill acquisition outcomes and show that both "pure" (training set by either firms or unions only) and "mixed" (training set by firms and unions) training scenarios may emerge at equilibrium. We show that firms have generally greater training incentives than unions, resulting in a higher product quality. In line with empirical evidence, we also find that the wage differential between high-skill workers and low-skill workers is lower when the training levels of the workforce are selected by unions than by firms. Finally, we analyze the optimal public training skill levels and demonstrate that both unions and firms under-invest in training in comparison with the social optimum. Yet, in this case the skill premium is the lowest.