I have only one experience in a tattoo parlor, but the tattoo artist had no problem using a customer's design. I got the feeling that this was common. They also had a book of standard designs, I didn't get the impression that the tattoo person cared at all what image was chosen.
We brought in a very good, clean copy of the design my friend wanted. They had a machine that copied it onto a special paper, and then they rubbed the image from that paper onto her skin.
Then they used a buzzing tattoo gun the do the actual tattoo. Once the image had been rubbed on her skin, things went pretty quickly.