Hi kyouell,
welcome to this Apple (user-to-user) Community!
Re: "Screen sharing during FaceTime call turned off my mic I initiated screen sharing when on a FaceTime video call on my MacBook Air (Sequoia 15.6.1) and I was automatically muted. The mute button didn't look "on" so I clicked it twice to turn it on and then back off again. This had no effect."
"How can I stop this from happening so I can share my screen and speak about what I'm sharing?:
Am no expert in this area, yet your question to the community deserves some response, if only to keep your post active, and provide more opportunity for experienced FaceTime users to share their knowledge with you.
Some things we may screen-share (eg: videos, apps we use for shareplay, etc.) may have audio of their own that needs the use of the mic. Sorry I don't know how much control we have over the mic during screen-sharing in FaceTime. (Am currently using other software and app for calls on my Mac)
Apple Support's articles may ormay not be helpful:
If needed: Choose a camera or microphone for FaceTime calls on Mac - Apple Support
However: Manage Screen Sharing and Screen Control in FaceTime - Apple Support
does not mention control of the mic.
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Re: "MacBook Air : (Sequoia 15.6.1):
Updating to the newest compatible MacOS version may (or may not) help with the issues you experienced
The latest MacOS Sequoia update is the third update since MacOS 15.6.1.:
What's new in the updates for macOS Sequoia - Apple Support
Eg: says: "macOS Sequoia 15.7.2
Linked article also says: "This update provides important security fixes and is recommended for all users".
.
NB: If you choose to update the MacOS: Please remember it is always wise to back-up your Mac first,
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Else: restarting your Mac (Apple menu > Restart) often restores order if something is not working as expected.
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Re: "The mute button didn't look "on" so I clicked it twice to turn it on and then back off again. This had no effect".
Wondering whether you checked whether the audio level during screen-share was loud enough for the task.
From the Apple Menu (In top menu bar) > System Settings > Sound > check sliders for volume, and balance.
For MacOS Sequoia:Change audio options for FaceTime calls on Mac - Apple Support has audio options.
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Resource: FaceTime User Guide For MacOS Tahoe 26
(or: for accurate info., choose your MacOS version at the top of articles.
Apple's "How to" articles can be reached from the "Table of Contents" on the Welcome page.):
FaceTime User Guide for Mac - Apple Support
All the best :-)