cursor action one machine does funny stuff on another machine

OK, this is weird. I have an iMac and a MacBook Air right next to each other. Both running Sonoma. When I move the iMac cursor all the way to one side of the screen or the other a vertical light blue bar appears on

that side of the iMac screen AND on the side of the MacBookAir screen. Or, the cursor on the Air disappears. That is, cursor motion on one machine does funny stuff on the other machine. What in the world is going on??? AirDrop & Handoff is turned off on both machines, so it's not that.