I would check Administration --> Issues --> Misc Schemes --> Permission Schemes --> Default Permission Scheme and see who you have set as "Assignable Users" under Issue Permissions.
It looks like the User has to be in Jira-Developer Group to be assignable not just a Jira-User. Once I added User to this Group I could assign a task even though the person is not a 'Developer'
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Do you mean that you are not seeing anyone in the assignee dropdown? Do they have Assignable permission set?
http://confluence.atlassian.com/display/JIRA/Managing+Project+Permissions
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