We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves analogously to classical Borel-Moore homology in the sense that there are proper push-forwards, cross products, and cup products with tropical cohomology classes, and that it satisfies identities like the projection formula and the Künneth theorem. Our framework allows for a natural definition of the tropical cycle class map, which we show to be a natural transformation. Finally, we characterize the rational polyhedral spaces that satisfy Poincaré-Verdier duality as those that are smooth.