When you first meet someone, you usually shake them ......... the hand.
This is a companion discussion topic for the original entry at https://english.best/questions/870,when-you-first-meet-someone-you-usually-shake-them-___-the-hand/
I don’t think shaking people by their hands is a good idea nowadays.
You can shake somebody by the hand or you can shake somebody’s hand, especially when you meet them for the first time, however, I found the following sentence in the Cambridge dictionary online:
The Princess was photographed shaking hands with AIDS victims.