The recent spread of mosquito-transmitted viruses and associated disease to the Americas motivates a new, data-driven evaluation of risk in temperate population centers. Temperate regions are generally expected to pose low risk for significant mosquito-borne disease, however, the spread of the Asian tiger mosquito (Aedes albopictus) across densely populated urban areas has established a new landscape of risk. We use a model informed by field data to assess the conditions likely to facilitate local transmission of chikungunya and Zika viruses from an infected traveler to Ae. albopictus and then to other humans in USA cities with variable human densities and seasonality. Mosquito-borne disease occurs when specific combinations of conditions maximize virus-to-mosquito and mosquito-to-human contact rates. We develop a mathematical model that captures the epidemiology and is informed by current data on vector ecology from urban sites. The model predicts that one of every two infectious travelers arriving at peak mosquito season could initiate local transmission and > 10% of the introductions could generate a disease outbreak of at least 100 people. Despite Ae. albopictus propensity for biting non-human vertebrates, we also demonstrate that local virus transmission and human outbreaks may occur when vectors feed from humans even just 40% of the time. This work demonstrates how a conditional series of non-average events can result in local arbovirus transmission and outbreaks of disease in humans, even in temperate cities.