The first one hands over the shift. the second one takes over the shift.

I work in factories a lot, and there are usually three parts to a shift change (though they are not always done) 1. Preparation. The person on the current shift prepares any information the person on the next shift might need and turns on/off any relevant devices if needed. 2. Handoff (or less often, handover): Before he/she leaves, the person on the first shift tells the next person anything they need to know that would be important in doing their job. 3. Takeover (or check-in): The person on the next shift formally takes over the duties. The person on the first shift leaves and has a good stiff drink and winds down.

There are many names; pass the torch, the baton, the shift and other names. I was a sailor and I think the idea of this came from ships because of how life is aboard ship. You have your watch, your station and your quarters. When it is your watch you are on watch and when your time on watch finishes you turn over the watch to the person or people replacing you. Many expressions came from sailing ships and the military. Good question.