Before sharing the document, run the Document Inspector and choose to clear personal information. Even though reviewer names will be visible during editing, once you save and close the file, that information will be cleared.
Here's how to run the Inspector: Make sure the relevant document is the active document in Word. Click File | Info | Check for Issues | Inspect Document. Clear all check boxes except for the one next to "Document Properties and Personal Information."
Click Inspect. Next to "Document Properties and Personal Information," click Remove All. Close the dialog box and save the file.
Stefan Blom, Microsoft Word MVP