Hand signing a print is a relatively recent thing, starting in the late 1800s. Original Rembrandt and Durer prints are not hand signed. Durer prints often have his monogram as part of the printed graphics.
In modern times, the artist’s hand signature on an original print shows that the print was personally approved as finished by the artist. The artist signs it when it’s all finished and meets his or her approval. Prints that didn’t come out right go unsigned and are often literally destroyed and tossed in the trash. This explains why art collectors pay more for a hand signed original print by a famous artist. The extra price is not just because it’s autographed, but because the autographed indicates the print was personally okayed by the artist.