Is there a way for users to download a copy of a SharePoint Excel 2010 file, save it locally, and have the local copy update with all future changes made to the SharePoint document? Or does anyone have any other suggestions for a single source of truth between files downloaded from SharePoint and the files saved on SharePoint? Additionally, we do not use OneDrive or Office 365.
With SharePoint 2013, you can use One Drive for business to share your libraries to locally. It is automatically update the SharePoint as well as Locally once a change has been made.
You can read this blog for more understanding.How to sync your SharePoint 2013 document libraries with OneDrive for Business
If you dont have the one drive for bussines then you can think about the Mapping the Drive.
- You can map your SharePoint library locally to your computer. this way you can manage the files easily. check this for more information.