What exactly is happening? Hers and His sound correct; but why?
"Send me a picture of hers" means to send me a picture that belong to her, or in which she was the photographer or artist (not sure if "picture" here means a photograph or a painting). "Send me a picture of her" means to send me a picture in which she is the subject, that is, she is the person (or one of the people) visible in the picture.
Similarly if you use a noun rather than a pronoun. "Send me a picture of Sally", I want a picture that shows what Sally looks like. "Send me a picture of Sally's", I want a picture that belongs to Sally or that was taken by Sally.
In context, "send me a picture of hers" could also mean that you want a picture that shows an object that belongs to her, or a person associated with her. It is short for "... of her X". Like, "I was thinking we could use Bob's house for the party, or we could use Sally's house." "I've seen Bob's house, but send me a picture of hers."