I've worked with these printers - at least the primitive, ancient neanderthal version of them.
Our machine would glue thin sheets of paper, layer by layer, and cut them with the outline of the part, plus a grid pattern on the non-part. After finishing all the layers, you'd pick off the non-part with a dental pick, leaving the part. The resulting model felt basically like wood. It was difficult to use for hollow objects because you could not get inside to pick out the hollow.
I would imagine that the newer printers would use some sort of polymer, and only lay it down where it needs to go. I'm sure that there would need to be some sort of support matrix, but it would be removable through some better method than picking.
We also had primitive programs that could turn a photo into a 3-d image. I'd discussed a business plan with a friend for making custom bobble-heads. You could send a picture and we'd create a bobble-head of whoever it was.
I think that 3-d mapping would be relatively easy now. I imagine a camera mounted like a microscope above a turn-table with precise measurements. Place the item, snap a few pictures as it is rotated, and the software triangulates the images and turns it into a 3-d image in the computer. That would be primitive, but would work. I'm sure you can get better accuracy using sonar or x-rays, or some such.
The knob that the guy made is nifty, but the same effect could be made using a block, a drill, and a woodruff key. Or, a set of vice-grips! Anybody ever seen the "There, I fixed it!" site?