As Steve stated, it is the sqlserver service account that needs to be granted write access to the path that you want the backup to be written to.
If you are creating a backup using UNC to your own computer,
be sure not to break the DRP for the db !
Check Books Online "Backup database" and use the keyword "for copy_only"
If I didn't have access to the db backup, I would ask one of the
sqlserver server admins to provide me a copy of the backup file, in
stead of taking the risk of being accused of data-theft !