No, falling on concrete is not necessary, provided you train with mats as a safety mechanism and you do not rely on mats to protect you from unrealistic techniques.
Use mats for extra safety for techniques that work without mats
Mats provide an extra margin of error while you are learning but should not be used to protect you from bad technique. When you fall, you should be executing proper falling technique by supporting your head while trying to reduce pressure, impact, and the energy absorbed by your body. If you train like this on mats, practicing these elements carries directly into falls on concrete.
Drop knee seoi nage is an example of a technique that only works with mats. The thrower in this technique drops their knees directly into the ground while attempting to throw. While this may not injure the thrower on a mat surface, this is totally inadvisable for the thrower on concrete. The mat is protecting against behavior that relies upon the ground being soft. This should be obvious from using your imagination; no one should need to test this on concrete.
You can choose to train techniques like this in the sport context, but personally I think the punishment your body takes is too much even with mat protection.
You can see how well a person falls on any flat surface
- Head hitting the ground is an automatic failure. Do not allow the person anywhere near hard surfaces
- Is the impact spread over a large area?
- Is the impact over the large area at the same time?
- Is force directed into the joints (for example, bracing with a straight arm? This one is bad.
- If the fall allows it, is energy directed away from impact? This could be rolling, where you preserve kinetic energy rather than take it as impact, or raising the legs during a rear breakfall to convert the energy of the fall into potential energy.
- If there is a roll, how much energy does the person need to return to standing? If they exert their legs significantly to get up, this tells you they have absorbed the potential energy in their body when falling. It is obviously preferable to avoid absorbing more energy.
Proper falling practice reduces forces regardless of surface
Regardless of training surface, you should be continually trying to improve your falls. You cannot restrict yourself to thinking, "I don't get hurt on the mat, so I won't get hurt on concrete." This is also a general problem with training mindless repetitions, if you never think about what you are doing, especially "the basics", you will not get better.
It may be useful to fall on hard surfaces occasionally to give yourself feedback, but this really should not be necessary. You can feel how you make contact with the ground on a mat surface and adjust to improve your falling technique to support your head, reduce pressure, or keep your frame from collapsing. You still get this sensory feedback while falling on mat surfaces. Did your head hit the ground? Are you concentrating force in your shoulder that makes you uncomfortable? Is a roll continuous? Unlike getting punched in the face, which is very different from not getting punched in the face, falling on mats is very similar to falling on concrete. If you compare two ways of falling on the mat, and one reduces the forces on your body, this will continue to be true on harder surfaces.
Good falling technique works on hard surfaces
It is definitely worthwhile to know you can take a controlled fall on concrete without getting hurt. Some people get the idea that mats are necessary to fall safely. Malicious opponents are, of course, a different story.
Yes, I have taken unplanned falls (solo) on concrete with only superficial injuries. Yes, you will get scrapes from little pebbles on the ground. No, falling on concrete beforehand was not necessary to prepare.