We investigate the direct-sum problem in the context of differentially private PAC learning: What is the sample complexity of solving learning tasks simultaneously under differential privacy, and how does this cost compare to that of solving learning tasks without privacy? In our setting, an individual example consists of a domain element labeled by unknown concepts . The goal of a multi-learner is to output hypotheses that generalize the input examples. Without concern for privacy, the sample complexity needed to simultaneously learn concepts is essentially the same as needed for learning a single concept. Under differential privacy, the basic strategy of learning each hypothesis independently yields sample complexity that grows polynomially with . For some concept classes, we give multi-learners that require fewer samples than the basic strategy. Unfortunately, however, we also give lower bounds showing that even for very simple concept classes, the sample cost of private multi-learning must grow polynomially in .