I didn’t mean to suggest that the inside surface was the only surface that should be sharpened, only that those methods were what I had seen for those surfaces because of the shape. Looking at several of the videos online there are a variety of sharpening methods used on the bevels including flat stones, abrasives on sticks and various strops. In any case, most of the sharpening is done on the bevels just as Sides stated. I am reluctant to admit it in this forum but I often power strop/hone the curved cutting edges of my tools as many carvers do.
Personally, I prefer gouges, carving the bowls before any other part of the spoon. I find bowls blanks easier to hold down while there are still plenty of flat surfaces and I can use both hands on the tools and use the power of the mallet.