We analyze data from the Fall 2020 pandemic response efforts at the University of Colorado Boulder (USA), where more than 72,500 saliva samples were tested for SARS-CoV-2 using quantitative RT-PCR. All samples were collected from individuals who reported no symptoms associated with COVID-19 on the day of collection. From these, 1,405 positive cases were identified. The distribution of viral loads within these asymptomatic individuals was indistinguishable from what has been previously reported in symptomatic individuals. Regardless of symptomatic status, approximately 50% of individuals who test positive for SARS-CoV-2 seem to be in non-infectious phases of the disease, based on having low viral loads in a range from which live virus has rarely been isolated. We find that, at any given time, just 2% of individuals carry 90% of the virions circulating within communities, serving as viral "super-carriers" and possibly also super-spreaders.