In acute cases the nasal swabs might be the best way to track the spread of the virus but I'm tending to think that faecal testing is required to isolate carriers.
https://www.cidrap.umn.edu/news-perspective/2020/04/study-many-asymptomatic-covid-19-cases-undetected"Questionable findings from single throat swab test
In the EID study, a 39-year-old hospital nephrologist who began having a dry cough on Jan 31 was hospitalized with fever on Feb 7 and diagnosed as having coronavirus on Feb 10. He lived with his wife, a lab physician with no patient contact; their 7-year-old twins with only family contact due to school closures; and the twins' retired 62-year-old grandfather and 64-year-old grandmother.
All family members were hospitalized on Feb 11 and remained asymptomatic throughout their 21-day stay.
All tested positive for COVID-19 except one who tested negative on four consecutive throat swabs but was positive on stool specimen testing and had high liver enzyme levels but no jaundice.
Another family member had a high D-dimer level, indicating blood clots. All abnormal lab findings normalized during the hospital stay. Three family members had abnormal chest computed tomography (CT) scans.
The wife, who had 11 serial throat swabs, showed negative results on two consecutive occasions and then reverted to a positive result. She also underwent serologic testing, which showed low levels of B lymphocytes but no coronavirus antibodies. The authors said that her case illustrates the challenges of interpreting the results of quantitative reverse transcription polymerase chain reaction (qRT-PCR) for COVID-19.
"Coronavirus Covid 19: NZ in pilot plan to sample sewage for virus"
"Recent studies have shown that live Sars-Cov-2 – the virus that causes Covid-19 – can be isolated from the faeces and urine of infected people, and can sometimes survive for up to several days after leaving the body."
The virus doesn't even have to be alive to be detected.