Yes, I narrowed down the problem further. It seems to be specific to the page restriction. All of my editors are restricted to be able to edit just one page. With this setting, they can edit their one page and upload new media (but unable to edit the metadata, which is my problem). When I turn this setting off (allowing them to edit any page), then the problem goes away and they can edit metadata.
To restrict their editing to just one page, I go to each user profile, and in the Posts/Pages/Custom Post Types Editor Restrictions section, I click the radio button for “Allow” and then enter the page ID number in the field called “with post ID (comma separated).”