If I understand your post correctly, the behavior you're describing suggests that the screen-sharing invitation may be initiating in the wrong direction. It could also be an issue with how the request is being handled between the two versions of macOS.
I suggest trying the following:
- On your Mac running macOS Golden Gate 27.2, open the Messages app.
- Select the conversation with the person using macOS Ventura.
- Click the FaceTime button at the top of the conversation.
- Select Ask to Share Screen, rather than Share My Screen.
- On the Ventura Mac, the other person should receive a request to share their screen. They can accept the invitation and choose whether to allow you to observe or control their screen.
If the invitation still behaves incorrectly, try initiating the connection directly from the Screen Sharing app:
- Open Screen Sharing, located in Applications > Utilities.
- Select New Connection.
- Enter the other person's Apple Account email address.
- Click Connect and have them accept the invitation on their Mac.
Apple provides additional instructions here: Share your screen using Messages on Mac
One important distinction: FaceTime's newer remote-control feature requires macOS Sonoma 14 or later on both Macs. However, traditional Mac-to-Mac screen sharing through Messages or the Screen Sharing app supports older macOS versions, including Ventura.