Thank you {{{{Mary}}}}
OK Inkedpalm (I'm may have to call you Inky, since I don't know your first name ;) ).
If you take a toothpick and drop a very small (verrryy small) drop of either lemon or lime juice (or straight white vinegar), onto the back of one of the pieces, and then look via the loupe -- you'll be able to see some fizzing or a few bubbles if it's coral, since this is the acid reacting with the shell and 'eating' away at it (don't worry, it's very minor).
Now you'll want to wipe it very well with a warm wet cloth, and dry it well -- to be sure to remove any/all traces of the acid.
You'll also get this reaction with other organic materials, but given what you have, it's one way to test. Another method, is to study the striations under the loupe. You'll often see faint streaks of irregularity in the color in some portions which is the 'grain' of the coral and sometimes a few pock marks (like tiny holes)
If you check the back, where the cut is flatter, you may be able to detect the open pores, depending on how the coral was cut.
If you don't see any of the above or it appears to be inconclusive, then by all means take the set to a jeweler. Most are gemologists, so they'll be able to view it under a higher powered lens to tell you for sure.
But it's like Linn says -- in all likelihood these are gemstones. Considering that it's been worked into a precious metal, I don't think that it could be otherwise.
