Who can make changes to the Backbone configurations?

Question

Cloud:

Immediate: (Project) admins on both sides can update the configuration.
With Review: (Project) admins on both sides can make updates to the configuration, but updates need to be approved by both sides before going live. Learn more about Review vs Immediate mode here.

Data Center:

HTTPS: (Project) admins on the initiated side of the sync can change the configuration.
Behind firewall: (Project) admins on both sides can change their part of the sync. They determine what goes out and comes in to their project.

In general, project admins can access the synchronizations in their projects. This can be further restricted per your requirements. Learn more here.