Can community users manage their own membership while logged in?

Our integrations don't naturally support communities. This is primarily because both the community and the licenses are very configurable.

That being said we do have customers that have told us they are using our product with communities. We have provided a Limited Access permission set which allows our triggers to fire and provides access to our objects and fields.

However, you will need to consider if your sharing model--depending on your community license type--will allow you to assign sharing access to the right records. Customer Community licenses, with their limited sharing sets, are likely not able to use any triggered features (field mappings to Mailchimp, including email changes).

The best advice we can give is to install our product in a sandbox which is fully featured and to test with your community and users to make sure it meets your needs. 

The data wizard can upsert members from Salesforce in the context of an internal user, so it may be a workaround that meets your needs. 

We are currently investigating various documentation and feature enhancements to support communities more. If you would like to be notified when that is live, please create a support ticket.