I am only speaking for boxing, which is the only sport I have experience with.
But if concussions and head injuries really worry you, don't spar and don't fight. You can hit the pads, work the bags, do the drills. But if you don't like the idea of getting hit in the head over and over, stay away from that ring.
Will you end up a dribbling mess if you spar a few times? No, of course not. Lots of people spar and have their fun - even log an amateur fight or two or five - without consequence.
But any time you fight, there's a real chance you could get fucked up and put down. It's the cost of doing business. This is the hurtin' game man. There's no way around that. And I know few people who were in this game for a long time that haven't seen some collateral damage by the time they're 40 or 50.
If you're not comfortable with that, no shame. Most people aren't. Hell, even I'm not entirely. But it's a seductive sport, and once you figure out that getting hit doesn't "hurt" like you think it does, it's easy to keep pushing a bit further each time. That's a choice only you can make.
Again, your mileage may vary. Some guys retire from long pro careers and are sharp until the day they die. Others get brain bleeds and die during an amateur fight. It's a total crap shoot.