As we observe a rapidly growing number of astrophysical transients, we learn more about the diverse host galaxy environments in which they occur. Host galaxy information can be used to purify samples of cosmological Type Ia supernovae, uncover the progenitor systems of individual classes, and facilitate low-latency follow-up of rare and peculiar explosions. In this work, we develop a novel data-driven methodology to simulate the time-domain sky that includes detailed modeling of the probability density function for multiple transient classes conditioned on host galaxy magnitudes, colours, star formation rates, and masses. We have designed these simulations to optimize photometric classification and analysis in upcoming large synoptic surveys. We integrate host galaxy information into the SNANA simulation framework to construct the Simulated Catalogue of Optical Transients and Correlated Hosts (SCOTCH), a publicly-available catalogue of 5 million idealized transient light curves in LSST passbands and their host galaxy properties over the redshift range 0 < 𝑧 < 3. This catalogue includes supernovae, tidal disruption events, kilonovae, and active galactic nuclei. Each light curve consists of true top-of-the-galaxy magnitudes sampled with high ( 2 day) cadence. In conjunction with SCOTCH, we also release an associated set of tutorials and the transient-specific libraries to enable simulations of arbitrary space-and ground-based surveys. Our methodology is being used to test critical science infrastructure in advance of surveys by the Vera C. Rubin Observatory and the Nancy G. Roman Space Telescope.