Each shared library has its own members and roles, which is what decides who can change what.
Managing members
Open Shared Libraries in the sidebar and open the library. Members and their roles are managed from there.
Roles run from viewer, who can use what is in the library, up to editor and manager, who can add content and manage membership. Your own role is shown next to the library in the list.
A lesson filed in a shared library can only be moved within that library, which stops content leaving a shared space by accident.