Do not disturb while screen sharing

### Feature Description

I would like to be able to configure to automatically have "Do not disturb" enabled while I am screen sharing.

### Use Case

While I'm sharing my screen, I would like to avoid notifications to appear which may contain sensitive information (i.e from chats).

Only the small dot on the notification widget should appear.

Perhaps, if possible, it would be neat to have the notifications popup when I stop screen sharing.

### Compositor(s)

- [x] All compositors
- [ ] Niri
- [ ] Hyprland
- [ ] MangoWC (dwl)
- [ ] Sway
- [ ] Other (specify below)

### Proposed Solution

A toggle under the "Do Not Disturb" section in DMS settings which reads something like "Automatically enable while screen sharing".

### Alternatives/Existing Solutions

Not sure if it's possible to implement, but Niri has the ability to block certain apps from screen sharing. If it was possible to block out the notification popups from screen sharing that would also be cool. Though I'm not sure if the other compositors have this ability, so a more universal approach would be to just (temporarily if possible) suppress the notifications.

No files, tests, or entry points are named. Start by tracing the DMS settings Do Not Disturb section and how screen-sharing state is exposed across the listed compositors; done means a toggle can suppress notification popups during sharing and restore them afterward, if supported.

Written by the indexing model from the issue text.