Do a tattoo artist have to draw the tattoo on a sheet of paper before placing the tattoo on your arm?

This does not always have to be the case, and in the instance of which you are speaking would be for an artist to draw you a custom design and then transferring it onto stencil paper. They could be using a stock piece of tattoo flash that they already have on file, then they simply have to run it through the thermofax machine with the stencil paper to make the stencil. They could draw the design right on your flesh with sterile skin scribe markers or Sharpie's, but these lines wipe off fairly easily, so a lot of artists don't prefer this method. Lastly, they could "free hand" the design right on you, which means that there is no stencil or sketch at all and the artist is creating the design and colors right out of his or her head.