what's stopping cheap, non-refugee laborers from scooping job postings from refugees? I don't think it would be possible to verify the refugee status of someone without leaking identifying info.

I think problem is solved by the fact if you aren't a refugee you are probably looking for work through normal channels. A job board like this is for people who are either doing something illegal (as a symptom of being a gov resistant platform), or who have no other options due to circumstances.