Recent analyses suggest that distance residuals measured from Type Ia supernovae (SNe Ia) are correlated with local host galaxy properties within a few kpc of the SN explosion. However, the well-established correlation with global host galaxy properties is nearly as significant, with a shift of 0.06 mag across a low to high mass boundary (the mass step). Here, with 273 SNe Ia at $z<0.1$, we investigate whether stellar masses and rest-frame $u-g$ colors of regions within 1.5 kpc of the SN Ia explosion site are significantly better correlated with SN distance measurements than global properties or properties measured at random locations in SN hosts. At $lesssim2sigma$ significance, local properties tend to correlate with distance residuals better than properties at random locations, though despite using the largest low-$z$ sample to date we cannot definitively prove that a local correlation is more significant than a random correlation. Our data hint that SNe observed by surveys that do not target a pre-selected set of galaxies may have a larger local mass step than SNe from surveys that do, an increase of $0.071pm0.036$ mag (2.0$sigma$). We find a $3sigma$ local mass step after global mass correction, evidence that SNe Ia should be corrected for their local mass, but we note that this effect is insignificant in the targeted low-$z$ sample. Only the local mass step remains significant at $>2sigma$ after global mass correction, and we conservatively estimate a systematic shift in H$_0$ measurements of -0.14 $textrm{km},textrm{s}^{-1}textrm{Mpc}^{-1}$ with an additional uncertainty of 0.14 $textrm{km},textrm{s}^{-1}textrm{Mpc}^{-1}$, $sim$10% of the present uncertainty.
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