QTP is able to save to shared network folders AND it does have the permission to do it.
The trick is to use the full UNC path. This means that if you wanted to save the file GlobalStore.tsr
inside of:
\\somewhere.corpnet.com\Groups\QA\Automation\Object Repository\
You would put in the 'File name' field the whole path:
\\somewhere.corpnet.com\Groups\QA\Automation\Object Repository\GlobalStore.tsr