Let's look at the CDC website,
https://www.cdc.gov/coronavirus/2019-ncov/vaccines/fully-vac...> Based on evolving evidence, CDC recommends fully vaccinated people get tested 5-7 days after close contact with a person with suspected or confirmed COVID-19.
For many people, this description could apply to daily activity, e.g. riding on public transit. So the CDC is recommended regular testing of vaccinated people. Since the vaccine can suppress symptoms, testing can be used to detect infection earlier, enabling isolation and/or treatments which only work early in the progression of Covid.
In a July 2021 paper from academics advising UK SAGE, https://www.gov.uk/government/publications/long-term-evoluti...
> Whilst we feel that current vaccines are excellent for reducing the risk of hospital admission and disease, we propose that research be focused on vaccines that also induce high and durable levels of mucosal immunity in order to reduce infection of and transmission from vaccinated individuals. This could also reduce the possibility of variant selection in vaccinated individuals.
Note the reference to transmission from vaccinated individuals. There was also a study of Vietnamese healthcare workers, showing that 70% of infections were pre-symptomatic, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3897733
> 69 healthcare workers were tested positive for SARS-CoV-2. 62 participated in the clinical study. 49 were (pre)symptomatic ... Breakthrough Delta variant infections are associated with high viral loads, prolonged PCR positivity, and low levels of vaccine-induced neutralizing antibodies, explaining the transmission between the vaccinated people. Physical distancing measures remain critical to reduce SARS-CoV-2 Delta variant transmission.