Hello,  I do not have such a perforator but we do have a CNC turret
punching machine for sheet metal.  It has lots punches and dies.
The punches get blunt in that their edges round off, and the same
happens on the dies.  We sharpen them by surface grinding the business
edge of the punches and the top of the dies.

For your perforator, I would proceed as follows:  Take the punch and
run it's rim (edge) into the top of your thumbnail at ~45 degrees,
like a chisel.  If it does not dig into the thumbnail but slides over,
then it is blunt and you need to sharpen it.

Stone the bottom edge of the punch on a diamond lap or sharpening stone
until the edge of it will dig into your nail.  You need to get it
perpendicular with the stone.  Make a jig to do it in the machine.

The same holds for the die but because the hole is too small for
testing with your nail, you have to look at it with a magnifier.  If
you can see any rounding at the rim/edge, you will need stone the die
down until the edge is sharp and crisp again.

It is not to hard to do.  You may be tempted to use a bench grinder or
finisher but I wouldn't because any heat may anneal the steel and then
it will just be too soft to keep an edge.

Best of luck and let me know how it turned out.

Best regards,
Bernt Damm
Sydney