In order for the user to edit the list you shared, they must complete the following steps:

  1. Go to Analysis > Manage Lists > Manage Basic and Advanced Lists
  2. Select List Admin
  3. Filter for your lists
  4. Click edit for the list

Alternatively, if the user does not have the platform manager role, or access to the List Admin, they can make a copy of the list you gave them access to and can save the copy as their own and make any edits as needed.