Absence of evidence is evidence of absence, from a strictly logical standpoint. How strong of evidence it is depends on how exhaustive the search is.
Usually, this argument comes up in nonexistence proofs. To prove something nonexistent, you need to demonstrate that it’s existence is impossible within the system you’re declaring it to be nonexistent in. This is often easy in mathematics, and sometimes even applies in the real world: we can say that a second Sol-sized sun in our inner Solar System is nonexistent because it would have to have certain physical effects that we can demonstrate are absent.
However, if mathematical, logical, or physical impossibility doesn’t apply (unicorns, supernatural anythings, aliens), then the only remaining absolute proof is to demonstrate exhaustive absence of evidence: if you’ve looked for “x” in all the possible places “x” might be and not found it, you’ve proven it’s nonexistence-using nothing but absence of evidence. On the other end of the scale, if you haven’t looked anywhere for “x”, then absence of evidence for “x” has no weight at all (I can’t even prove my roof exists without looking up).
Usually where the arguments start is evaluating how thorough the search has been. For example, I’d claim that the evidence for aliens not having visited us, abducted people, and left probes in their bodies is very, very strong based on having never found such a probe, even in people who claim to have them. A abductee believer would find all sorts of reasons why that’s logical, though, and claim that I haven’t searched nearly enough to make the statement. On the other hand, absence of evidence for the existence of aliens, anywhere, is meaningless, because we haven’t (and can’t, presently) search most of “everywhere.”
Given the relatively poor grasp many folks have of what constitutes evidence, valid scientific trial, and uses and abuses occam’s razor, arguments rarely get to this sort of subtlety effectively.