Functionally optimized networks abound in nature, efficiently and precisely controlling the propagation of inputs to perform specific tasks. The regulation of protein activity via allostery presents one of the most well-studied examples: such proteins utilize specific conformational or dynamical changes upon the binding of ligands to facilitate communication between distant active sites. Venation networks in animals, plants, fungi and slime molds also display a type of allosteric communication, having the ability to precisely distribute oxygen and nutrients from a limited number of inputs to locally support growth and activity. Whether via genetic evolution or dynamic adaptation, many of these networks are able to create and control allosteric functionality by locally tuning interactions between nodes. Taking inspiration from this ability to regulate function, we approach allostery as a problem in metamaterials design, asking whether it is possible to create synthetic mechanical and flow networks with allosteric properties. We show that not only is this possible, but is remarkably easy, only requiring a small percentage of interactions in a network to be tuned. Leveraging the large statistical ensembles of allosteric networks generated in this way, we show that the limits of multifinctionality in both flow and mechanical networks are governed by the same constraint satisfaction phase transition, unifying both systems into a single theoretical framework. Finally, we investigate the underlying mechanisms by which allosteric function is created in flow networks. We show that the relationship between structure and function in flow networks is topological in nature, not depending on local details of the network architecture. The approaches presented in this work for studying allostery in both flow and mechanical networks set the blueprint for understanding and controlling general functional complex networks.