Virtually all enzymes catalyse more than one reaction, a phenomenon known as enzyme promiscuity. It is unclear whether promiscuous enzymes are more often generalists that catalyse multiple reactions at similar rates or specialists that catalyse one reaction much more efficiently than other reactions. In addition, the factors that shape whether an enzyme evolves to be a generalist or a specialist are poorly understood. To address these questions, we follow a three-pronged approach. First, we examine the distribution of promiscuity in empirical enzymes reported in the BRENDA database. We find that the promiscuity distribution of empirical enzymes is bimodal. In other words, a large fraction of promiscuous enzymes are either generalists or specialists, with few intermediates. Second, we demonstrate that enzyme biophysics is not sufficient to explain this bimodal distribution. Third, we devise a constraint-based model of promiscuous enzymes undergoing duplication and facing selection pressures favouring subfunctionalization. The model posits the existence of constraints between the catalytic efficiencies of an enzyme for different reactions and is inspired by empirical case studies. The promiscuity distribution predicted by our constraint-based model is consistent with the empirical bimodal distribution. Our results suggest that subfunctionalization is possible and beneficial only in certain enzymes. Furthermore, the model predicts that conflicting constraints and selection pressures can cause promiscuous enzymes to enter a 'frustrated' state, in which competing interactions limit the specialisation of enzymes. We find that frustration can be both a driver and an inhibitor of enzyme evolution by duplication and subfunctionalization. In addition, our model predicts that frustration becomes more likely as enzymes catalyse more reactions, implying that natural selection may prefer catalytically simple enzymes. In sum, our results suggest that frustration may play an important role in enzyme evolution.