There are different kinds of corals growing on reefs in many places. IMHO the vast majority of collected corals come from various reefs in the South Pacific. Some corals are raised captively by growers, specialists and hobbyists.
The colors are mostly natural. Some corals will change colors based on the light they receive, just like we do. After a long winter, go stand in the summer sun and what happens? You turn either tan or red based on how much sun you get. Corals create pigments to protect themselves from light as well. They create different colors to protect themselves from different spectrums of light.
I collect some nice medium pea green zoas in the Florida Keys and have found that 2' deep in my DT they stay green. But on top of a rock island in the same tank where they are under only 4" of water, and therefore receive much more light, they turn a very pretty bright sky blue over a period of 2 to 3 months.