I saw an injury where someone shoved the tip of a mogito (or iaito if you prefer -- zinc/alloy bladed, unsharpened) in to his forearm and out by his elbow. Lots of stitches. Serious injury.
Another person I know, highly experienced, shoved a live blade through his own midsection on the side during a kata (obvious he made a mistake). Hospital trip.
And there is a famous event here in Southern California where a visiting sensei severed a finger or two during a live demonstration of kata.
And you don't want to know how many scars I have on my body from both training with the things but also as a full time sword polisher and craftsman. I walked in to the tip of a blade on a rack that was itself screwed to the workbench. I grabbed a tanto I dropped (hmmm, yeah, it's sharp). And I've simply slipped doing what I do. And the (what we call) deli slices I get on my hands sometimes are epic. And as one of the weirdest things was when I started having back problems and changed my position for polishing from the traditional posture I found that when I polished a certain part of the mune I had a habit of occasionally lifting a leg up (no longer locked down in the kamae) and sticking the tip in to my own leg right above my knee cap. It took three time in to exactly the same spot before I managed to break myself of the habit of absentmindedly lifting that leg a bit. Nothing worse than holding a blade embedded in your own leg, taking a deep breath, then pulling it out. Advice -- pull quick and don't flinch that leg muscle. Slow doesn't make it any better.
They are dangerous. There is no safety, no unloading. And people like to swing the things around even when they have no idea what they're doing (and frankly most in Aikido (with some significant exceptions) seriously have very, very little idea of how to handle a sword correctly).
That said... Just be safe. Don't do the things you don't know how to do. If you decide to try some stuff, well, best advice is to get some training. Otherwise go slow. I've seen guys on video saying "be careful" then promptly demonstrate cutting technique with a leading leg seriously close to being where it could take a major cut if they blew the cut they were doing. And then watching them cut ("Batter UP!") usually makes me cringe because it is an emergency room visit waiting to happen.
But again... They're only as unsafe as the person holding it. I have many friends and customers who don't train who have lovely swords in safes or on stands up and out of the way on display. They get them out, lovingly look at them, study them, enjoy the artistry (some can be fantastically beautiful), lovingly clean them, and then put them away. Folk can get really strident about saying "don't get a sword without sensei or training". For some who envision themselves as backyard samurai masters because they've seen *every* episode of Samurai Jack, sure, probably not a good idea. But I dare say most folk out there are reasonable, intelligent and fully mature enough to have a sword. Shrug.