You do not have multiple assignees. Jira has a single assignee, as multiple ones do not work in real life. The assignee is the one single person who should be responsible for the issue in its current state.
You can include other people as "interested" or "the backup responsible person" etc. Use the watchers field possibly, or maybe a group-picker if you know it's going to be a member of a particular group, but most use a custom field of a user picker.
Or create sub-tasks, assigned to the people who are responsible for doing their parts of the task.
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