Absolute not. There is no way for the tattoo ink to enter your bloodstream unless you are jamming something all the way through your skin and reaching the veins beneath. A tattoo machine, even if setup to run at its longest stroke length will not travel far enough to go completely through your skin. Unless you're working on a scar tissue area, that is a different ball game.
Someone stated that when you get a tattoo, ink is placed between the first and second layer. Well, they are not completely correct. The ink settles in the dermis which is the second group layer of skin. This is after the needles carrying the ink pass through all '5' layers of the epidermis. And different areas of the body have different thicknesses of skin. That is why using an artist working from home or even an artist in a shop that just knows how to tattoo but does not actually know about the canvas they are working on can cause damage.