"Assign" a ticket simply means making clear who has the responsibility for the next steps. JSM makes that clear to a customer with the "Waiting for Customer" status.
The Assignee field is for licensed agents only, and then, by default, only people in the "Service Desk Team" project role. You don't need or want to have customers in this field, because they don't need or have to know your service management processes and procedures.
How I've worked around this is use the "Approvals" feature:
This way, they will get a "Notify approvers" email when a issue is "assigned" to them. If you don't use approvals otherwise, you can customise the text to outline this process. Hope that helps!
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