Thanks for raising this question around the visibility of the Members Only group. For an external community, presuming guest access is enabled and the guest users (unauthenticated visitors) are able to access the root/home page, then both Open and Members Only groups will be visible to guest users. The difference here is that Members Only groups require you to join the group before you're able to create and reply to content, whereas an Open social group is available to all users with access to the site.
If your intention is to require authentication for all users to the entire site, then you'll need to disable guest access entirely. If your intention is limited to restricting access to specific Members Only social groups, then this gets a bit tricky. Unfortunately, there's no way to explicitly restrict guest access on an externally accessible site to specific social groups, so it's either all or none. This is configured via the permissions in the Jive Admin Console under Permissions > Social Group Permissions, and this is outlined here:
You'll want to locate the "Everyone" permission group, as this applies to guest users and anyone not already specified, and revoke the "View social group" permission level. Authenticated users will fall under the "All Registered Users" permission group. As long as you have a separate setting for the "All Registered Users" group (this is the default), then your authenticated users should remain unaffected.
Feel free to test this out and let me know if you have any other questions. If you require more in-depth assistance, feel free to submit a support case in your Jive Customer Group and a team member will be happy to assist you. If you're not familiar with the support process, please check with your community manager for assistance.
Thank you Jeph that was helpful and I appreciate the detail! I will work with my community manager to see what he wants to do to fit my needs.