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Edit/View checklist by assignee only
This guide works only in Jira Classic projects. Next-Gen projects do not give the option to grant permissions for dynamically changing user fields like assignee or reporter.
If you want to configure permissions so that only the Jira issue assignee can edit the checklist (toggle, edit, etc.), then please follow the steps below:
Enable custom work on checklist permissions on the Permissions page.
Configure Jira project permissions so the selected permission is granted to "Current Assignee" only