Content Pages are a new type of page that have the peculiarity that they include fragments and content within the fragments. Because of this, when publishing a page of this type from staging to live it is desired to also publish all the fragments it includes along with their configuration and content.
It is worth noting that even though content pages are frequently created from page templates, it is not currently desired to have staging of page templates. This means that with local staging, there should be only one set of page templates independently of whether staging is enabled or not. And when creating a new page, that set of page templates should be listed as available independently of whether the page is being created in live (when staging is disabled) or staging (when it's enabled).
A similar approach should be used for fragments. That is, there is only one set of fragments and there is no need to publish them from staging to live.
When using remote staging, the fragments and page templates would exist only in the staging server. When publishing a content page to the live server, only that page along with its configured fragments and their content (FragmentLinks) would be published.