You can only assign a task that appears in a shared List. There are two ways to do this:

If a task is not in a shared list

Tap on the task to open the detailed view. Then tap on the three dots on the right upper corner, then on the Move icon and choose a shared List. A box with a person’s name (usually you) will appear under the ‘Reminder’. Tap that option and assign the task to anyone included in that List.

From scratch

Create a List. Share the List with the person you want to assign the task to. Create the task inside that List. Learn more on our Lists Help Guide.

Make sure you attribute a date to that task, otherwise it will stay in the List, but it won’t appear on the assignee’s Today screen at any point if there’s no specific day for the task to be shown.