Yes, it requires a single trick in database and some modifications on code. Look at FS db structure, To copy task into private project, you have to change the project_id field from non-kitchen to kitchen and (optional) make realation between them. Remember that first field is auto_increment so you have to put NULL there.
INSERT INTO flyspray_tasks SELECT NULL, <ID of kitchen project here>, task_type, date_opened, opened_by, is_closed, date_closed, <all of table fields here...> WHERE task_id=<id of previously opened task>
should do the trick, but I'm not an expert though.