The Zwicky Transient Facility (ZTF) is performing a three-day cadence survey of the visible northern sky (similar to 3 pi) with newly found transient candidates announced via public alerts. The ZTF Bright Transient Survey (BTS) is a large spectroscopic campaign to complement the photometric survey. BTS endeavors to spectroscopically classify all extragalactic transients with m(peak) <= 18.5 mag in either the g(ZTF) or r(ZTF) filters, and publicly announce said classifications. BTS discoveries are predominantly supernovae (SNe), making this the largest flux-limited SN survey to date. Here we present a catalog of 761 SNe, classified during the first nine months of ZTF (2018 April 1-2018 December 31). We report BTS SN redshifts from SN template matching and spectroscopic host-galaxy redshifts when available. We analyze the redshift completeness of local galaxy catalogs, the redshift completeness fraction (RCF; the ratio of SN host galaxies with known spectroscopic redshift prior to SN discovery to the total number of SN hosts). Of the 512 host galaxies with SNe Ia, 227 had previously known spectroscopic redshifts, yielding an RCF estimate of 44% 4%. The RCF decreases with increasing distance and decreasing galaxy luminosity (for z < 0.05, or similar to 200 Mpc, RCF 0.6). Prospects for dramatically increasing the RCF are limited to new multifiber spectroscopic instruments or wide-field narrowband surveys. Existing galaxy redshift catalogs are only similar to 50% complete at r 16.9 mag. Pushing this limit several magnitudes deeper will pay huge dividends when searching for electromagnetic counterparts to gravitational wave events or sources of ultra-high-energy cosmic rays or neutrinos.