You can assign a checklist item to any user who has the Assignable user permission (for more info, see the Jira documentation). Once a user is assigned to an item, their avatar will appear on the right side of the item row.
To assign an item:
Hover over an item in the checklist and click the Assign User button.
Click Assign to me to assign the item to yourself, or click in the Assign to drop-down to assign the item to another user.
Viewing your assigned items
Users do not currently receive notifications when they are assigned to items.