The recent emergence of divergent SARS-CoV-2 lineages has raised concerns about the role of selection within individual hosts in propagating novel variants. Of particular concern are variants associated with immune escape and/or enhanced transmissibility. Though growing evidence suggests that novel variants can arise during prolonged infections, most infections are acute. Understanding the extent to which variants emerge and transmit among acutely infected hosts is therefore critical for predicting the pace at which variants resistant to vaccines or conferring increased transmissibility might emerge in the majority of SARS-CoV-2 infections. To characterize how within-host diversity is generated and propagated, we combine extensive laboratory and bioinformatic controls with metrics of within- and between-host diversity to 133 SARS-CoV-2 genomes from acutely infected individuals. We find that within-host diversity during acute infection is low and transmission bottlenecks are narrow, with very few viruses founding most infections. Within-host variants are rarely transmitted, even among individuals within the same household. Accordingly, we also find that within-host variants are rarely detected along phylogenetically linked infections in the broader community. Together, these findings suggest that efficient selection and transmission of novel SARS-CoV-2 variants is unlikely during typical, acute infection.