Hi Marty,
What's your detailed Outlook 2016 version number? Please click File > Account > About Outlook to collect the detailed version.
I followed your steps and was able to reproduce your issue with Outlook 2016 version 16.0.7927.1024.

If I opened the sharing request by double-clicking, nothing happens when I clicked "Allow" or "Deny". As a workaround, please try to click the Allow and
Deny button in email Reading Pane to send response:

Regards,
Winnie Liang
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