Assuming a 'new' medalion with the post un-cut and it has the blind hole in the end.
To back up the face of the medalion, use a piece of steel or brass rod the dia of the face or just slightly less than that.
Cover the face of this rod with a piece of thin leather. the thinner the better. It's just to protect the details on the face of the medalion when you rivit the post .
Too much leather thickness and you create a soft, spongy surface and then you have to pound harder than necessary to rivit the post on the back.
The post should have a shallow blind hole in it's center. That's for the quick and simple riviting process.
You use a hammer but not directly on that post.
You place a steel 'center' like a center punch or even an old lathe center into that hole. Tap the center with the hammer and it will mushroom out that post evenly all around and down onto the washer.
The face of the medalion is placed down onto the leather faced rod during this rivit process.
It can be a 2 person thing if you can't juggle the grip, hammer and center punch at one time. But with some practice it becomes easy to do as the force to actually do the riviting and the fact that only one wack is generally needed makes the whole thing quite easy.
I've been told by others that they do the process in a bench vise. The grip with medallion & washer in place betw the open jaws.
A short backer for the medallion as above and a short pointed 'center' set in place against the end of the post.
Closing the vise jaws carefully and watching the post mushroom/rivit as the center is pushed into the end of the post does the job.
I've never done them this way, . It must work OK,,but lots of stuff to juggle & hold in place while closing the vise up!
You could probably do the operation on a simple single stage reloading press as well with a couple simple home-made tools like the backer and a center to fit in the press.
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Re-setting used medallions where the posts have been ground off when removing them,,I just clean up the ends of the posts and then epoxy the medallions and washers back into place. Once set-up, you can usually add a few rivit strikes to the end of the post for a more orig look if that's needed.