How to separate the two parts linked by these hinges?
You might be able to remove the hinge pin by gently tapping from the bottom up, using an appropriate size punch and hammer.  The photos don't show the top and the bottom of your hinges well.
If the hinge pin is flanged at both ends (like a nail head), then you won't be able to do this, though.

Otherwise, you'll have to remove the rivets...
You could try drilling them out, but take care not to damage the hinge and/or the mirror frame.

Likewise, you could take a Dremel cutting wheel and carefully cut off the rivet crimps (See the image below).

But, for small jobs like this and with small soft-ish rivets, I've had good luck just mashing the rivet-crimps with vise-grips and/or diagonal cutters -- squeezing it small enough that the rivet could be pushed through the whole.  If the rivets are not too tough, this can be the fastest and cleanest way to get them off.




Note for completeness: In some cases, rivet crimps and/or heads can be chiseled off, but in this case, it would probably wreck your cabinet in the process.