Community testing programs focused on the unvaccinated population are being enacted in populations with mixed vaccination status to mitigate SARS-CoV-2 spread. Presumably, these policies assume that the unvaccinated are driving transmission, though it is not well understood how viral spread occurs in mixed-status populations. Here, we analyze a model of transmission in which a variable fraction of the population is vaccinated, with unvaccinated individuals proactively screened for infection. By exploring a range of transmission rates, vaccine effectiveness (VE) scenarios, and rates of prior infection, this analysis reveals principles of viral spread in communities of mixed vaccination status, with implications for screening policies. As vaccination rates increase, the proportion of transmission driven by the unvaccinated population decreases, such that most community spread is driven by breakthrough infections once vaccine coverage exceeds 55% (omicron) or 80% (delta), with additional variation dependent on waning or boosted VE. More broadly, the potential impacts of unvaccinated-only screening fall into three distinct parameter regions: (I) "flattening the curve" with little impact on cumulative infections, (II) effectively suppressing transmission, and (III) negligible impact because herd immunity is reached without screening. By evaluating a wide range of scenarios, this work finds broadly that effective mitigation of SARS-CoV-2 transmission by unvaccinated-only screening is highly dependent on vaccination rate, population-level immunity, screening compliance, and vaccine effectiveness against the current variant.