The dynamics of SARS-CoV-2 replication and shedding in humans remain poorly understood. We captured the dynamics of infectious virus and viral RNA shedding during acute infection through daily longitudinal sampling of 60 individuals for up to 14 days. By fitting mechanistic models, we directly estimate viral reproduction and clearance rates, and overall infectiousness for each individual. Significant person-to-person variation in infectious virus shedding suggests that individual-level heterogeneity in viral dynamics contributes to superspreading. Viral genome load often peaked days earlier in saliva than in nasal swabs, indicating strong compartmentalization and suggesting that saliva may serve as a superior sampling site for early detection of infection. Viral loads and clearance kinetics of B.1.1.7 and non-B.1.1.7 viruses in nasal swabs were indistinguishable, however B.1.1.7 exhibited a significantly slower pre-peak growth rate in saliva. These results provide a high-resolution portrait of SARS-CoV-2 infection dynamics and implicate individual-level heterogeneity in infectiousness in superspreading.