Let me first confirm your question: there is an additional authentication issue when you open a file downloaded from SharePoint with a local app. Please correct me if I have misunderstood.
If I understand correctly, please kindly provide more information below to let us work further:
**1.**I guess the issue you described should be related to the incorrect credentials on users’ local computers. Given this situation, please try the following methods on one affected computer first to check if there is any improvement.
- Open Word on an affected PC: Windows key + R >Input WinWord and press Enter to open the Word client>Open a blank document and Click File tab on the upper left corner>Click Account>Sign out all the accounts.
- Remove all the stored credentials in the credentials manager (Control Panel > User Accounts > Credential Manager > Windows Credentials). (NOTE: This will remove your stored passwords.)
- Sign into Office programs, such as Word with your SharePoint credential.
- Open downloaded documents to check if the same issue occurs.
**2.**If the issue persists, please try to open the document in app on SharePoint Online. Test whether additional authentication will be required in this case.
**3.**SharePoint Online has an information rights management (IRM) for downloaded files. This policy requires an E5 license. You can contact your external SharePoint administrator to see if IRM is turned on for the entire tenant and applied to the document library where your downloaded files are located. Check to see if there is an expiration time set for access rights after the file is downloaded. It is likely that the access rights to the file you downloaded expired.
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