Heat engines and information engines have each historically served as motivating examples for the development of thermodynamics. While these two types of systems are typically thought of as two separate kinds of machines, recent empirical studies of specific systems have hinted at possible connections between the two. Inspired by molecular machines in the cellular environment, which in many cases have separate components in contact with distinct sources of fluctuations, we study bipartite heat engines. We show that a bipartite heat engine can only produce net output work by acting as an information engine. Conversely, information engines can only extract more work than the work consumed to power them if they have access to different sources of fluctuations, i.e., act as heat engines. We illustrate these findings through an analogy to economics alongside analytically tractable model systems including a cyclically controlled 2D ideal gas, a Brownian-gyrator heat engine, and a quantum-dot information engine. Our results suggest design principles for both heat engines and information engines at the nanoscale, and ultimately imply constraints on how free-energy transduction is carried out in biological molecular machines; we highlight this application by inferring the magnitude of information flow inside photosystem II.