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Some dogs can communicate such things. Bunny (of What About Bunny) can vaguely, imprecisely, communicate "some kind of pain somewhere" using a button-activated buzzer system. She can sometimes name the approximate location of the pain after a minute or two of thought. (See https://redirect.invidious.io/watch?v=RN_ZpyS6Fkc&t=34 )

I have no idea how you'd get "I'm in pain" from the associated body language; but, then again, I'm not a dog. In this case there were behavioural cues, but I don't know how I'd tell if there weren't.

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The usual way we get pain from horses is because they start to do things differently than they usually do. Not having a normal "walk cycle" (slight pain is easiest to see in trot), scratching bellies more than usual, rolling more than usual, rolling less than usual, eating slower than usual, eating faster than usual, shitting less than usual, shitting more than usual; you get the idea. O11y is not that bad — but you must have a "known good" baseline.

For toothache specifically, we'd do a differential check: does the horse eagerly eat a soft mash, but won't eat hard cubes? If so, time to grab that tongue and do a visual inspection (and if everything looks normal, it's still worth palpating).

(a typical vet complaint is that owners don't do any of the above, and first notice only when stench ultimately makes it obvious to check for toothache)