How do I get fields attached to a user to display to anonymous viewers?
After poking around on this for a while I discovered that it is definitely an organic groups permission issue. For the record, I have the 7.x-1.5 version of OG installed.

In OG global permissions (admin/config/group/permissions) the fields attached to the user (and nothing else) are all listed under Organic Groups Field Access along with fields attached to groups and fields attached to group content types. When I give access to non-members the fields appear for anonymous.

I suppose this functionality is designed so you can limit user access to people who are connected through a group. This isn't entirely senseless, but can be frustrating if you don't anticipate that control over some objects will be trumped by OG permissions.