I believe the commentary that tells you how much damage, if you/your opponent was stunned, etc. should show the amount of damage your opponent takes from poison/fire since we only know these effects last anywhere from 1-4 turns, but can't quite tell how long the effect lasts if we're the ones doing the poisoning/setting on fire.

I just look at their HP bar.

Usually I do as well but when I actually land an attack with poison effect, I can't really tell if it was still there or not since both damages are there and my short-sighted self doesn't keep track of health very well. Also in cases where you do more damage than the opponent has health, you can't tell either (though you probably wouldn't need to in that case 'cause the opponent's already defeated)

Yeah, you usually halve the opponents HP with poison, so I look to see if the bar is creeping down or not when my attacks are dodged. But a more precise indication wld be nice.