In my estimation, they should not be visible. At least as far as the backscatter X-ray machines go, they are terrible at distinguishing objects held flush to the skin (including, ironically, explosives). For something pasted on the skin, or in this case under it, the only possibility to distinguish it from the skin itself would be if it were comprised of a high quantity of heavy elements, metals and such. The radiation that an object is bombarded with is absorbed and scattered differently to the extent that the atoms comprising it are different.
I really have no clear idea what is in each particular tattoo ink. But unless the ink is comprised of massive quantities of heavy metals (unlikely), it should not react any differently to X-ray radiation than the skin it's embedded in, thus making it "invisible"
I know less about how millimeter waves rebound off different materials, but in as much as their primary means of detection is just bouncing off a surface, I think tattoos should be invisible to them too.