During submonolayer homoepitaxy, instability in the shapes of growing two-dimensional islands can develop due to the diffusion-limited aggregation of deposited adatoms at their edges. However, in metal (100) systems, periphery diffusion is typically efficient, quenching this shape instability, and resulting in simple near-square or near-rectangular shapes of isolated islands. Despite this feature, growth coalescence shapes resulting from collision of two or more growing islands are nontrivial. These coalescence shapes are elucidated here by developing three complementary formulations: (i) suitable atomistic lattice-gas models analyzed by kinetic Monte Carlo simulation; (ii) deterministic rate equations for the dynamics of kinks along island step edges; and (iii) continuum theories for step-edge evolution. Characterization of coalescence shapes is important as they affect interlayer transport during multilayer growth. Such a characterization is also necessary to enable coarse-grained modeling of film growth with a realistic treatment of the evolution of island edge morphologies, e.g., using level-set methods.