I've only ever received a stone from a wild pet battle, when I captured a Snowy Owl. (level 16) Can't remember the secondaries.
But I've also been battling non-stop lately, and have received nothing for defeating the opponents. Logically
, I would think you could get a stone from defeats; the per-requisite for capturing one seems a bit odd, because that would rely on players having enough space to capture (even though the limit is 650 now).
But it's not happened to me yet, so I can't really say.
I hope people screenshot some looting of stones, might clear things up.