Yes, the hosted Files folder is a shared folder on the Network.  All users within your organization will export to your specific Site ID folder, but you can of course export to your own specific folder within your Site ID.  
You cannot access other organizations files, just as other organizations cannot access your files. 

If you wish to export to a "secure" Files Folder (for more sensitive documents), please see BB229574.
For more information on how to use the Blackbaud Files folder, see BB172995