macOS has Facetime to make video calls. Messages is for texting. But you can't share your screen in Facetime which is only possible in Messages. Do I need to understand why?
There is a nice document from Apple at https://support.apple.com/guide/messages/screen-sharing-icht11883/11.0/mac/10.13. It talks about a Screen Share button. Where is that? I need screenshots - a text only explanation doesn't work for me.
Here is a short introduction into screensharing with Messages.
Step 1: Make sure that the person you want to do screensharing with is in your sidebar. Here I'm talking to myself:
Step 2: Select Buddies -> Invite to Share My Screen or Ask to Share Screen.
If both menuitems are disabled then screensharing is not possible. Here I've been talking to the iPhone of the Favourite Sister:
Step 3: I selected Ask to Share Screen because I wanted to see the screen of my laptop from my iMac.
The Screen Sharing app opens in the menubar and I can see it connecting:
￼Step 4: On the other side my laptop gets a notification about the screensharing. The computer is in German. The screenshot says "xxx wants to share your screen" and "accept/deny".
After clicking on accept you will get another window. You have the options to operate the screen or to view the screen. Click on the Accept button to allow the screensharing.
And the result is that I can now see the screen of my laptop: