We present a study of the effective (half-light) radii and other structural properties of a systematically selected sample of young, massive star clusters (>=5x10^3^M_{sun}_ and <=200Myr) in two nearby spiral galaxies, NGC 628 and NGC 1313. We use Hubble Space Telescope (HST) WFC3/UVIS and archival ACS/WFC data obtained by the Legacy Extragalactic UV Survey (LEGUS), an HST Treasury Program. We measure effective radii with GALFIT, a two- dimensional image-fitting package, and with a new technique to estimate effective radii from the concentration index of observed clusters. The distribution of effective radii from both techniques spans ~0.5-10pc and peaks at 2-3pc for both galaxies. We find slight positive correlations between effective radius and cluster age in both galaxies, but no significant relationship between effective radius and galactocentric distance. Clusters in NGC 1313 display a mild increase in effective radius with cluster mass, but the trend disappears when the sample is divided into age bins. We show that the vast majority of the clusters in both galaxies are much older than their dynamical times, suggesting they are gravitationally bound objects. We find that about half of the clusters in NGC 628 are underfilling their Roche lobes, based on their Jacobi radii. Our results suggest that the young, massive clusters in NGC 628 and NGC 1313 are expanding, due to stellar mass loss or two-body relaxation, and are not significantly influenced by the tidal fields of their host galaxies.