I watch combat sports that involve striking and the vast majority of the time they get hit they don't show any sign that it hurts. What kind of pain are they feeling? I would think that considering how hard pros can punch and kick that getting hit would be extremely painful and cause serious injuries.
In my experience, there are a few reasons why professional/trained fighters do not feel pain as much as a normal person.
Higher pain tolerance
They are constantly taking hits and their bodies have become accustomed to pain. Fighters know in which situations pain is unavoidable and they may have developed a different response to it. Also, by mentally anticipating pain it becomes easier to deal with.
Over time, fighters develop hardened bodies through striking and blocking. Conditioning exercises help to strengthen your body in certain areas. For example, some fighters take blows to their abdominal muscles to develop a higher resistance to pain in that area. Kung fu fighters have been known to thrust their hands in hot sand to toughen them. While these may seem somewhat extreme, through sparring regularly you are effectively conditioning your body and developing a higher resistance to pain.
Adrenaline, also known as epinephrine, can enable a fighter to feel less pain when it is pumping through their blood. The initial pain is dulled as the body uses the adrenaline rush to protect itself. But with a severe injury, like a broken bone, the pain will always catch up to you eventually!
Epinephrine is known to dull pain for a time thus allowing someone not to feel as much pain as they actually have. However, it is temporary. I suspect, but have no evidence, that this is what allows competitors to take blows and not register the pain till later on. A lot of training require "soaking" of blows thus both reducing the energy dumped into the body by parry/deflection and getting adapted to the pain of being hit.
Your body turns off you ability to feel pain when you're in a serious fight. I once fought with a broken collar bone. It was a bit tender, but I didn't feel any pain until the following morning. And for me getting hit hard in the face was just a bright explosion of light and a moment of disorientation. If it was a particularly hard strike, you may feel the same type of pain as bumping your head against a door or something but it's not traumatic. You just don't really feel pain. The shock of getting hit is worse than the actual hit.
The important thing is that there are two phenomena limiting your performance:
Physical limits -- injury or pain so severe that you CANNOT continue.
Psychological limits -- a combination of fear, surprise, stress, or self-preservation instincts make you REFUSE to continue (including fleeing, surrendering, collapsing, dialing back your own intensity, etc.) long before there are injuries that truly, physically prevent you from performing. I'm not being judgmental here... it is these instincts that help you avoid sustaining severe injuries most of the time.
For somebody who is not used to being hit hard, the pain threshold of #2 is MUCH LOWER than the actual physical limits of #1. Through practice -- simple acclimation to being in circumstances where you take hard hits -- you can raise the threshold of pain that you are willing to endure psychologically.
In other words, with practice in painful/stressful/adrenalized situations, you can quite often endure hits (that are very painful but short of severe physical injury) that somebody unpracticed would not tolerate.