Yes. In your project permission scheme, grant the "Browse permission" only to reporter and a developer group or user.
You can also grant the same permission to a custom field and in that case the value will change for every issue. i.e. you can have a separate developer for each issue.
Ah, um, there's two problems here.
If you grant permission to "Reporter", then it means anyone with "create" permission will see the issue. It doesn't work the way it sounds. Jira actually thinks "Reporter" means "Anyone who MIGHT report an issue", not "The person who ACTUALLY reported this issue". There is a fix for this - there's a "reporter browse" permission which is disabled by default (because it can lock a system if used incorrectly, so Atlassian force you to read the docs before using it)
Second, David said "open issues". That changes the question from standard permissions to status-dependent permissions. Which Jira doesn't really do.
Good point about the reporter. If you need the person who created it, as opposed to reporter, you will have to capture the creator in a user custom field with the help of a workflow post function.
I missed the status specific permission part. But JIRA does support status specific permissions using workflow properties. http://www.j-tricks.com/1/post/2011/02/permissions-based-on-workflow-status.html
You could set up a issue security level (by default) where only Reporter and your developer group has access to this issue. For details see https://confluence.atlassian.com/display/JIRA/Configuring+Issue-level+Security
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