How do I export a membership list for a private group?

There are a couple of options

An automation, which is probably not an option but I'll briefly cover it 

Triggered manually via the automation sandbox
Passing in the group id and sending the results via email, we have used this approach for other ad-hoc extracts in the past such as listing digest subscriptions
In this case the automation would simply need to iterate through the group members 
$core_v2_groupUserMember.ListEffectiveGroupMembers($groupId, "%{ PageIndex = 0, PageSize = 500 }"))

Alternatively

You could use a REST endpoint to list the users, this would need some knowledge of a tool such as PostMan and for the Auth keys to be setup in the community but once you have this setup it would open up the whole world of REST APIs to extract data 

community.telligent.com/.../list-group-user-rest-endpoint

There are a couple of options

An automation, which is probably not an option but I'll briefly cover it 

Triggered manually via the automation sandbox
Passing in the group id and sending the results via email, we have used this approach for other ad-hoc extracts in the past such as listing digest subscriptions
In this case the automation would simply need to iterate through the group members 
$core_v2_groupUserMember.ListEffectiveGroupMembers($groupId, "%{ PageIndex = 0, PageSize = 500 }"))

Alternatively

You could use a REST endpoint to list the users, this would need some knowledge of a tool such as PostMan and for the Auth keys to be setup in the community but once you have this setup it would open up the whole world of REST APIs to extract data 

community.telligent.com/.../list-group-user-rest-endpoint