We measure the propagation of spatially localized spin waves in NiFe thin films through local inductive detection of the dynamic magnetization. A pulsed magnetic field excites a linear superposition of spin wave modes with a distribution that is predominantly driven by the spatial dependence of the in-plane excitation field. The results of numerical micromagnetic calculations exhibit excellent agreement with experiment and show that a comprehensive account of spatial nonuniformity and propagation is necessary to accurately measure the intrinsic damping rate.