We investigate the direct-sum problem in the context of differentially private PAC learning: What is the sample complexity of solving $k$ learning tasks simultaneously under differential privacy, and how does this cost compare to that of solving $k$ learning tasks without privacy? In our setting, an individual example consists of a domain element $x$ labeled by $k$ unknown concepts $(c_1,\ldots,c_k)$. The goal of a multi-learner is to output $k$ hypotheses $(h_1,\ldots,h_k)$ that generalize the input examples. Without concern for privacy, the sample complexity needed to simultaneously learn $k$ 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 $k$. 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 $k$.