Agree what stays on the shared surface
Keep equipment used by both people in a stable place, such as the monitor, keyboard, and suitable charging setup. Give each person their own tray or folder for active work. A desk that only becomes usable after someone sorts another person's private papers is not really ready to hand over.
Write a short end-of-turn routine
Make the handoff short enough to do after an ordinary workday. If both people need the desk at once, agree on time blocks or a second working spot rather than expecting a tray to solve a scheduling problem. Keep private documents out of the other person's workspace.
The finishing person can save their work, remove private documents, return the chair or screen to an agreed neutral position if needed, and leave the shared equipment ready. Ask the other user what they need at the start of their next turn. Keep the routine short enough for an ordinary busy day.
Separate a space problem from a time problem
If both people need the desk at the same hour, rearranging the drawers will not solve it. Put time blocks or a second workable location on the calendar, and allow setup time between them. If only one person needs privacy for a call, let that requirement shape the schedule without revealing details of the conversation.