There are at least two ways you can finish inside and outside surfaces of the ring. Generally, the annealing process will leave a certain amount of colored residue on the outside surface of the coin. Depending on what the base metal is, the color of this "jewelers scale" will range from dark grey (almost black) to a lighter range of grey or possibly a brown color. In either case, you may finish the coin so that much of the background retains the darker color; or alternatively you can choose to polish the entire surface to a mirror finish. The choice is yours.
If you leave a lot of the color in the background of the coin, the details of the original engraving will be more prominent. To acheive this finish you need to first lightly sand only the tops of the highlights with a very fine grit sand paper or emery paper. Then apply some metal polish to the ring and polish until the highlights and edges are quite shiny and prominent. This leaves the background darker.
Alternatively if you want to go for the mirror finish; use a fine steel wool and polish the entire surface of the coin, thereby removing much of the background color as well as polishing the highlights. Then apply some metal polish and buff the ring until it reaches a mirror finish. One of the most popular metal polish products is called Mothers Mag & Aluminum Polish. It is usually available anywhere auto supplies are sold.
The details on the inside of the coin ring may be finished in the same manner as those on the outside of the coin.
There is also something known as a Jewelers Polishing Cloth. This cloth is actually a two-part item where one half of the cloth is impregnated with a metal polish, while the other half is merely a buffing cloth. The shine of any ring can be restored with this type of cloth.
This completes this series of postings. Bear in mind, I have iterated only the very basic steps and most coin ring craftsmen will use them as a start and then modify and alter them to form their own techniques and methods.
Also, there are several good videos on YouTube that describe this basic process or some form of it. I would recommend if you are just starting, to watch one or more of the available videos. Here are a couple that will get you started. My coin rings are available at Ross Coin Rings or Ross Coin Rings Etsy Shop