How can I learn to comfortably share responsibility with my baby's caregiver?

First... ...When you ascertain that the caregiver is in fact worthy of trust. Second, when you start trusting yourself to still be important to your child even if the caregiver does some things better.
When you are ready. It is natural for a mother to be protective of their child. Even though you picked a caregiver/nanny for your family it is still going to be a struggle to completely let go. You should take it in the stages that you are comfortable with.