I would think that if a third of the population has a tattoo, we'd actually know with almost certainty what the effects are.
> FDA considers the inks used in intradermal tattoos, including permanent makeup, to be cosmetics. When we identify a safety problem associated with a cosmetic, including a tattoo ink, we investigate and take action, as appropriate, to prevent consumer illness or injury. The pigments used in the inks are color additives, which are subject to premarket approval under the Federal Food, Drug, and Cosmetic Act. However, because of other competing public health priorities and a previous lack of evidence of safety problems specifically associated with these pigments, FDA traditionally has not exercised regulatory authority for color additives on the pigments used in tattoo inks. The actual practice of tattooing is regulated by local jurisdictions.
As regards long term population studies... we'd also need to know if what's in tattoo inks has changed over time, which seems plausible?
[1] https://www.fda.gov/cosmetics/cosmetic-products/tattoos-perm...
I wouldn't count on it. It took us 100 years to find out smoking was extremely bad, even though the majority of the population was doing it.