You don't have to dismiss or snooze the same notification on every machine in Notezilla . If the computer where the note is dismissed, syncs the notes first with the cloud then other computers will automatically cancel the active reminder when they sync.
If you dismiss on Computer 1, then that computer must sync the notes. Only then the cloud will be aware of the dismissed note. So now when Computer 2 & 3 sync, it will receive the dismissed note and cancel all active reminders.
Learn more about the automatic sync that happens every minute through this short video: