Most carvers will sign their name on the side of a finished seal. Many will add the date and a bit more information. Some will use a variety of calligraphic styles and carve decorations into the stone. Depending on the color of the stone you may want to fill in these lines with paint. No matter what color the stone is, any lines you cut into it will usually show as white. On a dark stone this is fine but on a light-colored stone your side inscription may not be visible. In this case you can use a little bit of black paint and dab it into the lines and then wipe off the excess from the surface of the stone. Ink can be used but there’s a tendency for ink to soak into the stone and stain it (this can happen with paint too). You can also use white paint on a dark stone and some carvers use gold paint, though I consider this a bit gaudy. There’s no reason you can’t use other colors if you like, but black and white provide good contrast without attracting too much attention to themselves. Be sure to brush any dust out of the side carving before applying the paint, as it will not stick otherwise.
The illustrated seal is by Wang Shishen (1686-1762) with a side inscription filled in with black ink. From Sun Weizu, “The History and Art of Chinese Seals” p.434.