I believe all users in your organization would have access to the Assets core service. I can't find any specific permission that needs to be assigned in order to access the interface. I believe that edit permission will be limited to the uploader or system administrator. The main documentation for Assets can be found here.
I'm trying to connect with a counterpart that is more familiar with Assets on this, but If there is something specific you'd like me to test, l can do so in my sandbox environment.