The emergence of divergent SARS-CoV-2 lineages has raised concern that novel variants eliciting immune escape or the ability to displace circulating lineages could emerge within individual hosts. Though growing evidence suggests that novel variants arise during prolonged infections, most infections are acute. Understanding how efficiently variants emerge and transmit among acutely-infected hosts is therefore critical for predicting the pace of long-term SARS-CoV-2 evolution. 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 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, and are rarely detected along phylogenetically linked infections in the broader community. These findings suggest that most variation generated within-host is lost during transmission.