Check your Project's "Permission Scheme" and see if what accounts/groups/roles are configured in the "Assignable User" permission. You also need to make sure that that account also have the "Browse Projects" permission so that he can view your project issues.
Good point on the permissions - the project could well be limited to just yourself or unassigned. I don't think the "browse" is right though - the user is saying that they can't find anyone in the assignee field for the issue - that means they're looking at the issue and trying to assign it, so they must already have browse and assign rights on it. If you meant that the potential assignee needs to be able to see the issue, it's a bit unfriendly, but you can assign issues to people who can't see them.
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