Even if the ALT text field is completed when the image is uploaded, each time I'm trying to use that image I get a message that the ALT text is required to be completed. I'm using Sitecore 8.2. Any ideas?
Sorry but with a fresh installation of 8.2 I am reproducing your steps, and there is no issue. Two things you may want to consider:
1) Since the Alt field is not shared, are you looking at a different language version?
2) When looking at the properties of an image the alternate text field will be empty, but under that it will state default alternate text with what you entered.