Are tattoos and permanent make-up safe to use?

Yes, but... Tattoos are generally safe, but some people are allergic to the elements that are in the tattoo dyes. Unfortunately, when that happens, other than injecting steroids in the affected areas, and ultimate tattoo removal, the inflammation will be chronic. Another problem is that over time, the tattoo dye migrate, so the lines becomes fuzzier. This can be problematic with tattoo brows and eyeliner...
Probably yes. Unless you are allergic to one of the dyes that are used for the person that does the procedure does not use sterile instruments.