Coins do have a mint finish. They have a certain luster to them. The bullion or coin proofs are struck twice for that mirror finish. If you ask a coin collector, if a human finger touches the coin, the finish is ruined ?
Pure silver bullion or round (private mintage) or pure silver coin (government backed like the silver eagle) will get an orange peel finish on the surface if they are stretched too far or reduced too small. This is a finish that cannot be removed. If you do not take it to this point, the finish stays very close to the original but the annealing process changes the crystalline structure within the matrix of the metal. It will change the look of the mint luster or proof finish.
For finishing, I like to use sanding and buffing blocks.
Here is the one I use from Sally’s beauty supply:
It is strange but I like them. My mother was a manicurist so a lot of these types of things were laying around. I have tried traditional jewelers tools for this but I like these for finishing. My friend Joshua Janes found this particular buffing block.
On the detail sides of the coin, I use this very fine grey abrasive side to remove the fire scale so the black max can patina the silver most efficiently. I then patina and buff the patina with the white side of the block. This seems to “set” the patina a bit better. This does not remove anything, just buffs to an extreme high finish. I then come back and remove the patina carefully from the high points only on the detail and then buff every thing again with the white side. I do the same on the inside of the ring. I remove the sides of the block and stick them together so it will fit inside the ring. 
I come from a blacksmith/Damascus knife maker background. I learned for a mirror finish on metal, you want to double your abrasive fineness each time and alternate the direction 90 degrees each time you go to a finer abrasive and remove the marks from the last abrasive. This technique dates back to the Japanese sword polishers that would achieve a perfect polish.
With the cut edge of the coin, I like to use a deburring tool first then remove those marks with a #4 file. Remove those marks with a flap sander 150 – 200 grit. remove those marks with a 3M, soft abrasive wheel. Then remove those marks with a rouge buffing wheel to achieve a mirror finish on the cut edge and inside reed edge. I can also finish the edge with ought electricity by using the purple and yellow sanding blocks from Sally’s. 
For the detail finish, less is more but you need to remove that fire scale from the anneal. That’s why I like the buffing block. Final finish includes a light buff with a polishing cloth and Renaissance wax on the outside.
There are many ways to skin a cat so its good to find a process that works for you. I usually adapt my method over time as well.
Some of the prettiest coins I think are:
  • walking liberty half dollar
  • Morgan dollar
  • Columbian Exposition half dollar
  • Isle of man Angel
  • Japanese silver coins
  • Swiss silver coins
  • Indian silver rupees
  • Tunisian coins
  • Ukrainian coins
  • Platinum eagle
  • gold eagles
  • Swiss silver coins