IANAA, but I imagine that they can detect the source of the waves and check electromagnetic observations from the same area. BH+BH = no electromagnetic radiation, star+star = electromagnetic radiation from 2 stars.
Of course, it's not easy to detect such fine measurements and correlate between different instruments.
Of course, this can be corroborated with electromagnetic observations. But, to my knowledge, not all NS-NS mergers are witnessed in telescopes for various technical limitations.
I think they know the difference just by looking at the mass. i.e. we think it is hard to form black holes smaller than ~3.3ish solar masses and we don't think neutron stars can be more massive than 2.2 solar masses.
This is why we get articles like [1] where there is an issue when we think we've found something between those numbers.
And yes, the foolproof way of checking whether a NS was involved is to follow up with telescopes. But the constraints on position from GW aren't always good and so you can't always find it.
[1] https://www.sciencemag.org/news/2020/06/gravitational-waves-...
https://twitter.com/di_goldene_pave/status/13223229779841433...