NOTE: Before you start, to be on the safe side, it is recommended that you make a backup copy of your Word document.
Click File | Info
Next to Document Properties and Personal Information click the Remove All button (as shown above)
Your personal data has been removed from the Word document. Save as usual, you are finished.
Just for your reference, the following shows and example of the File | Info screen BEFORE and AFTER removing the personal information on a document.
You can see that the modified dates and names have been removed.
I used Word 2013. I've seen it's also possible in Word 2010.