In Zoom there, is no need to "pass the ball" or "make someone presenter" in order for them to share their screen.
By default, any participant can Share Screen if no one else is sharing. The Host can overwrite when someone else is sharing.
If you like to have a different Screen Share controls (click ^ next to Share Screen):
Select who can share in your meeting and if you want only the host or any participant to be able to start a new share when someone is sharing.