A test with high sensitivity will have a low false positive rate.
A test with high specificity will have a low false negative.
So having a test with high sensitivity but low specificity will result in trust in the positives, but not trust in the negatives?
A sensitive test will catch many positive cases. It may or may not have false positives though, eg a test that’s always gives the right answer vs a test that always returns positive no matter what.
A specific test will give you few positive results when the true answer is negative. You could use it to rule something out. One test might say “patient has A or B condition”, and a second test with high specificity may then rule out A or B, leaving B or A, respectively, as the probable condition.
That would mean that while you catch most cases of the virus, you’d also get a bunch of false positives. Flipping a coin as a hypothetical test would give false positives and negatives, or low specificity and low sensitivity.