What does it feel like when you get hit - ie., punched in the face or take a kick to the leg, etc.? I've only ever been hit while wearing protective gear (in Academy).
To lean on someone else's words for a moment:
Getting hit in the head doesn't hurt, but it'll knock you out. Getting hit in the body hurts, but it won't knock you out.
-- paraphrased from Sean O'Grady, former World Boxing Association Lightweight Champion
The simple answer is that getting hit hurts, but how much it hurts is almost entirely dependent upon the pain tolerance of the recipient. One of the things that occurs during regular training and sparring is that the trainee both increases fitness, which offers some protection against abuses to the body, and increases tolerance for discomfort. What initially may seem painful later seems merely uncomfortable, and as one trains long enough, against others of similarly increasing skill level, the ability to strike efficiently increases, which means that more power can be delivered with less effort than when training began. But that doesn't mean that training partners and sparring partners lighten up as they advance. My experience was that we all kept hitting each other harder because we were tolerating it. That said, we still weren't blasting one another full power with everything we threw lest there be too many split-open heads to deal with, or too many broken ribs, fingers, toes, mashed noses, torn shoulders, wrecked knees, and so on...
For instance: guy kicked my eye socket with his heel, splitting the skin just under the eye where the skin overlaps a boney ridge. The blow itself didn't seem all that uncomfortable, really, a good solid sock, sure, but I've hurt worse bumping my head into the pointy corner of a kitchen cabinet. In any event, I kept fighting, but the instructors stopped the session because, unbeknownst to me, my face was bleeding and my eye was swelling up. It turned an amazing rainbow of colors for a week or two before it finally opened up again and I could see.
Another for instance: guy kicked my ribs and busted at least one of them. I didn't know that he'd broken any, only that he'd made contact, and that it was uncomfortable. So I kept fighting and knocked him out of the game with an uppercut. Day later, he looked fine, but I hurt every time I breathed. (This, by the way, lasted a few weeks.)
Another for instance: grappling with a guy, I put him in an Achilles lock, he put me in a heel hook and tweaked it. Just a little. Instructor made us stop. My crappy lock aggravated my opponent with discomfort (this is supposed to be a pain-compliance lock: next time you feel like propping your feet up, put the Achilles tendon -- right aboutish behind your ankle, at the bend -- on the sharp edge of a table, then apply pressure to imitate the feel), but his little tweak tore a few little bits in my knee. Result, he was walking around fine the next day, but I was limpy for many. The thing is, this particular lock doesn't hurt like the other one, but it starts wrecking things quick.
The leg kicks in Thai boxing hurt piles and piles when shins clack on the sharp ridge (or against knees or elbows), but don't hurt so much when contacting on the flat part. The more banged up they get over time, the less sensitive they become (or the less sensitive to them Thai boxers get).
But the worst dang-it-all pain came from stick fighting. Specifically, a little thing called a thumb pinch. Egad, I hated that little trick. Maybe I have more nerves in my thumbs than everyone else, but it seems less a bash-hurt-go-numb thing and more a grinding-this-isn't-going-to-stop thing. It's performed when one kali guy (eskrima/arnis) pinches the middle joint of his opponent's thumb between two sticks. Since both fighters are armed with sticks in fists, the setup is halfway there. I can't imagine ever being clever enough to slip one of those on someone in a fight, but it was danged uncomfortable to learn.