Sandstone, quartz and obsidian are all pieces I've experimented with. I live in the desert southwest of the U.S., and most of these are plentiful here. Sandstone varies greatly - a lot of it I've tried is much too coarse and blocky/blunt in grit shape and it'll do more damage than good, by burring or heavy rolling of the edge, without much real abrasion. And the quartz here is very hard, but also much too coarse in grit size to be very useful. And true obsidian is 'amorphous' in nature, meaning it has no discernible crystal structure. This means it's generally not hard enough, or strong enough microscopically, to abrade anything very hard. I tried quartz and obsidian samples in a 'scratch test' on glass (simple plate glass), comparing each of those to using the corner of a black hard Arkansas stone to do the same. The quartz easily scratched the glass, as did the Arkansas stone. But the obsidian couldn't scratch the glass at all - it was too soft, even in relation to the glass, which itself isn't very hard.
I did get lucky finding one piece of very fine-grained siltstone (or a very, very fine sandstone), which behaved much like a waterstone when used wet. The grit shed easily from the piece, like a very muddy waterstone. I used it to regrind an entirely new edge on a Case folder's blade in 420HC (HRC 55-57). It actually worked pretty well for that, and I used some of the shed 'mud' from the stone as a compound on some leather scrap to strop the burrs away. That's the ONLY piece I've had impressive results with, in all of the looking I've done. It was a small piece, just fitting into the palm of my hand. I've wished I could find a bigger slab of that stuff.