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We present optical and near-infrared ($ugriYJH$) photometry of host galaxies of Type Ia supernovae (SN~Ia) observed by the \textit{Carnegie Supernova Project-I}. We determine host galaxy stellar masses and, for the first time, study their correlation with SN~Ia standardized luminosity across optical and near-infrared ($uBgVriYJH$) bands. In the individual bands, we find that SNe~Ia are more luminous in more massive hosts with luminosity offsets ranging between $-0.07 \pm0.03$ mag to $-0.15\pm0.04$ mag after light-curve standardization. The slope of the SN~Ia Hubble residual-host mass relation is negative across all $uBgVriYJH$ bands with values ranging between $-0.036\pm 0.025$ mag/dex to $-0.097\pm 0.027$ mag/dex -- implying that SNe~Ia in more massive galaxies are brighter than expected. The near-constant observed correlations across optical and near-infrared bands indicate that dust may not play a significant role in the observed luminosity offset--host mass correlation. We measure projected separations between SNe~Ia and their host centers, and find that SNe~Ia that explode beyond a projected 10 kpc have a $\rm 30\% \ to \ 50\%$ reduction of the dispersion in Hubble residuals across all bands -- making them a more uniform subset of SNe~Ia. Dust in host galaxies, peculiar velocities of nearby SN~Ia, or a combination of both may drive this result as the color excesses of SNe~Ia beyond 10 kpc are found to be generally lower than those interior, but there is also a diminishing trend of the dispersion as we exclude nearby events. We do not find that SN~Ia average luminosity varies significantly when they are grouped in various host morphological types. Host galaxy data from this work will be useful, in conjunction with future high-redshift samples, in constraining cosmological parameters.