As a tattoo artist, I would say don't always trust every tattoo artist to know everything. There are some people who stumbled into their carreers in this field and don't know much about technique let alone the chemical compositions of the inks they use.What Are Tattoo Inks? | Tattoo Reaction to MRI
Having said that, I find very few people are allergic to pigments, but the ones who are are often allergic to red. These people are often allergic to other things containing red such as licorice, red candy, etc. I've had a dermatologist explain it to me as the phenol thaylene content in red pigments. Some people have just a mild reaction such as small bumps and redness; others will have the red in the tattoo scab violently and their body will actually push the pigment out over the long healing process. I've seen this happen and the tattoo might look perfectly fine except all the red is gone! Anyway, even a pretty well informed tattoo artist might not know the chemical breakdown of all their inks. So know what things you are allergic to before getting the tattoo, or let the artist know before you plan to go ahead.