It depends on how you set it up, and there are two layers here.
First, look at the project permission scheme. This says things like "Issue Edit: Reporter, Assignee and members of the role called developer". It sounds to me like you need to go over the permissions and check they do what they need. Specifically the "edit" permission here - you probably want that to say just "assignee"
Secondly, status - this is NOT a field, it is just a display telling you where an issue is. You can NOT edit status, as there's nothing there to edit. You change status by moving issues through a workflow. The workflow transitions (the action of moving from status to status) have things called "conditions", which are rules on who can perform the transitions. So, for example, a transtion from "ready to test" to "being tested" might have the condition "only people in the QA group can do this"
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