I have installed decorative stone in some of my walls, I really like how the look like, though the pictures don't make justice, but I'm not really happy with the edges since I don't have a wall that covers the sides. The sides look like the stones have just been cut or put in there like a 3D wallpaper.
I'd install some wooden trim / decorative moulding and paint it to match nearby woodwork.
Make the left vertical edge of the decorative-stone even by cutting where needed (e.g. middle row of upper photo).
tack/glue appropriate sized stripwood behind front edge of moulding as shim.
Attach the shimmed moulding using adhesive (as presumably used for the decorative-stone)
(I'd also grout all those stepped joints)
Another option is to continue the decorative-stone a short way around the corner - some manufacturers of these types of panels allow for this. (random example)
I don't particularly like resolving things like this with YAM (Yet Another Molding). It tends to make a nice, subdued edge more pronounced and conspicuous. You have a very nice line there now that fits well with the stone style. The perfectly straight line of wood trim doesn't fit well, to my mind.
I'd mix up some flat gray paint that coordinates well with your stone color and simply paint the edges. If you find the right color, they'll probably almost disappear. You could also use a slightly lighter or darker color and sponge on a bit of texture to make it match even better.