Hello @frenetixx
When you are the host of a meeting if you go to the little arrow button on the sharing key you can choose who can share in your meeting by choosing advanced sharing options. I will provide some screenshots to assist you from an active meeting of mine to help you.

click on the acorn (arrow) next to the green sharing button.

then choose who can share in your meeting.
Lets say that you just want that one person to share in your meeting and no one else. Then go to the participants button and then select "more" next to that participants name -> then make that person a co-host of the meeting. This will allow that participant to be the only other participant to share.
For more information on sharing please see https://support.zoom.us/hc/en-us/articles/201362153-Sharing-your-screen-or-desktop-on-Zoom
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