In order for the user to edit the list you shared, they must complete the following steps:
- Go to Analysis > Manage Lists > Manage Basic and Advanced Lists
- Select List Admin
- Filter for your lists
- 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.