You can export issues, if you go to the Issue Navigator, type a JQL query and then use the Export option. You can do it without admin permissions.
But you can import without admin permissions only if you are on Cloud. The Import option is available for non-admins in Cloud. You can find more info here:
In this post, we’re celebrating users who have gotten five accepted answers on the Atlassian Community. These folks are not only fabulous and helpful humans, they are our future Community Leaders! ...
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