This doesn't look like a hipchat question - is your tag wrong?
Anyway, no, not off the shelf, but yes, there's an easy fix if you're not using OnDemand.
The problem you've run into is a common one - the natural thought is that "browse = reporter" means "the person who reported this issue can see it". But Jira doesn't see it that way, it thinks "anyone who might have reported this issue can see it". There are reasons for this, but they're not very interesting so I won't bore you.
There is a simple fix - there's a "reporter browser" permission which you can enable and use instead, and it works perfectly. Caveat - it can break Jira if you use it outside the "browse" permission, which is why it's disabled by default. See https://confluence.atlassian.com/display/JIRA/Current+Reporter+Browse+Project+Permission
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