Problem
Templates and Modules need to be shared between wikis. It was requested ages ago, but mostly due to technical limitations (cache invalidation), it has not been implemented yet.
Proposal
Until it gets solved in a more fundamental way, we can already introduce a very easy solution that can be done without any changes to the server-side code. This way users can start experimenting with it, build useful templates, and figure out the social aspect of this feature and its limitations. And developers will see what functionality is actually needed for this to work well, and have an easy migration path once that functionality is in place. To do this, we introduce a bot that copies any pages marked with a {{AutoSync}} from the central wiki (Meta - which is the central OAuth wiki - or Commons, MW.org, etc) to any wiki listed in the corresponding wikidata item, and only if the target also has an {{AutoSync}} template. This approach solves the biggest issue - stale cache invalidation, because copying a template automatically triggers dependent page refresh on the local wiki.
For example, Template:GraphChart on commons would have a template at the top {{AutoSync}}, which would add the page to Category:AutoSync, and also show a message explaining what auto-sync is. It will also transclude template documentation from the doc sub-pages, e.g. /doc/en. Lastly, the GraphChart template will also be added to the Wikidata, where users will list all wikis that should receive a copy of this template. The auto-sync bot will go through the Category:AutoSync, and copy GraphChart to all those wikis. The bot will also copy documentation in the proper language as a /doc page, and insert an appropriate info text at the top, explaining that this page should not be edited because it will be overwritten.
How to add a new target
If a user decides to add a new wiki to the list or the template recipients, they would need to create a new page in the target wiki, containing only some magic template like {{AutoSync}} (or the local language variant). Once the page is created, they will also need to add that page to Wikidata as a local version of the original template. On the next bot run, the template will be auto-synced to the new page.
Sync pausing
If at some point users want to stop or pause page cloning for a local wiki, they can edit the page and change the info template at the top - adding a parameter "pause=explanation" - in which case the infobox will change and will contain a warning that this page is not being synced at the moment because of the given reason.
Permissions
Lastly, the bot could automatically set certain edit permissions on a template page in case there are too many pages that use it to prevent large-scale page regeneration vandalism.