If you overhear someone, in the privacy of their house, threatening to murder someone and go to the police, surely you don't expect this report being thrown out and you being yourself charged with the violation of someone's privacy instead?

IIRC if the evidence wasn't lawfully gathered (which it sounds like it was, tbh) then it wouldn't be a mistrial, it would be thrown out and then the prosecution wouldn't have any evidence of any crime.