It is always my preference to guide teachers in the direction of teaching their elementary school students not to reveal personally identifying information on classroom blogs. That said, I also recognize that sometimes the only way a teacher will try blogging is if they can restrict access to just students and their parents. The directions in the video below cover the steps you can take to restrict access to Edublogs, KidBlog, and Blogger blog posts. Edublogs and KidBlog run on WordPress. If you're using a WordPress blog, follow the steps outlined for restricting access to Edublogs posts.
You can find screenshots of the process here.