I'm wondering if anyone else has had this experience.
I've got a connection to some servers via Remote Desktop and RD Tabs (both RDP connection) from my machine. If I use Teams to screen share with another user they can click around and interact with the RDP sessions but cannot type into it. Their keyboard is seemingly unresponsive. However if I/they click onto another window that's not RDP then the keyboard works fine.
Not helping much, but we have similar problems. The request control button does not work if the person you are trying to control is on a RDS Teams environment. Seems like those features are not perfectly implemented yet.