The answer depends on whether you are using Lightroom Classic or Lightroom. Because you posted in the Lightroom Classic community, I assumed you are using Classic. Below are the answers for both applications.
If you do not want a photo to be synced to the Adobe cloud:
- For Lightroom Classic, do not include the photo in any collection that has sync enabled. This is easy in Classic because Classic uses local storage as primary storage, so nothing in Classic is synced by default. Photos are synced only if you manually enabled syncing for a collection.
- For Lightroom, do not add the photo to Lightroom. This is because all photos added to Lightroom are synced to the Cloud, because Lightroom uses the cloud as primary storage. You cannot have photos in Lightroom that are not synced to the cloud.
If your concern is that you do not want photos to be public, you can still use syncing in both Classic and non-Classic. In both Classic or non-Classic, uploading does not automatically make photos public. The only photos that are public are those in a collection (Classic) or album (non-Classic) that you specifically both Share and make public. So if your concern is about the public seeing your photos, you can sync photos and while keeping them private. They are private until you make them public.
Thanks, it was definitely helpful!