I can't try it out myself anymore, so I can only guess what might be the reason.

Do non-admin group members see the group description (as opposed to non-group members who are not admins)?

Do you use any 3rd party plugins that might add additional functionality to groups or might alter the Elgg core access level handling?

If you can see the description as admin it means that any access restrictions are simply overriden by the admin privileges. Now you need to find out what exactly restricts the access/visibility for non-admins. The could be due to some settings of Elgg for the corresponding groups and/or by some options added by some additional plugins. I can only suggest to think a bit about what plugins are possible candidates for altering the visibility and look through all relevant site / plugin / group settings. Maybe you come accross this one option that causes the issue.