Share and Publish - new Google Sites
Publish your site to a specific audience, your organisation or publicly and see the changes since the last publish
Publishing - Capabilities
Publish/share with a specific audiences
Specific people for an intranet (or for all your Google Workspace users)
Publicly for a public website (or anyone with the link)
See the changes since the last publish and turn on/off this option
Change published URL
Allow site to appear in search results
Work on site does not affect published version
Copy link to published site
Unpublish your site at any time
Review changes and publish feature was released to rapid release on 04 December 2019 on a 15-day rollout and scheduled release on 14 January 2020 (again 15-day roll-out period).
Publish a Google Site to a Secondary Domain
To map a secondary domain you need to create a site with a user account that uses the secondary domain as its sign in domain. This allows you to publish the site to the secondary domain and then map the domain. For a demo see Publish to and Map a Secondary Google Workspace Custom DomainGoogle Workspace.
Publishing - Limitations
Respects Google Workspace Drive Publish permission - your Google Workspace admin can prevent public publishing for example or external sharing (this is not an option for personal Google Sites)
Cannot publish individual pages or a subset of the site - the Review changes and publish feature informs you of what has changed since the lat publish, so this will help you decide if it's the right time to publish or what changes need to happen before publishing.