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We investigate the spatial structure and evolution of star formation and the interstellar medium (ISM) in interacting galaxies. We use an extensive suite of parsec-scale galaxy merger simulations (stellar mass ratio = 2.5:1), which employs the "Feedback In Realistic Environments-" model (fire-2). This framework resolves star formation, feedback processes, and the multi-phase structure of the ISM. We focus on the galaxy-pair stages of interaction. We find that close encounters substantially augment cool (HI) and cold-dense (H2) gas budgets, elevating the formation of new stars as a result. This enhancement is centrally-concentrated for the secondary galaxy, and more radially extended for the primary. This behaviour is weakly dependent on orbital geometry. We also find that galaxies with elevated global star formation rate (SFR) experience intense nuclear SFR enhancement, driven by high levels of either star formation efficiency (SFE) or available cold-dense gas fuel. Galaxies with suppressed global SFR also contain a nuclear cold-dense gas reservoir, but low SFE levels diminish SFR in the central region. Concretely, in the majority of cases, SFR-enhancement in the central kiloparsec is fuel-driven (55% for the secondary, 71% for the primary) -- whilst central SFR-suppression is efficiency-driven (91% for the secondary, 97% for the primary). Our numerical predictions underscore the need of substantially larger, and/or merger-dedicated, spatially-resolved galaxy surveys -- capable of examining vast and diverse samples of interacting systems -- coupled with multi-wavelength campaigns aimed to capture their internal ISM structure.