The term local refers to the location of the shared object. In this case, Adobe Flash Player stores the SharedObject file locally in the client's home directory.
When you create a shared object, Flash Player creates a new directory for the application and domain. It also creates an empty *.sol file that stores the SharedObject data. The default location of this file is a subdirectory of the user's home directory. On Windows, this directory is the following by default:c:/Documents and Settings/username/user_domain/Application Data/Macromedia/Flash Player/web_domain/path_to_application/ApplicationName/objectName.sol
If you request an application named MyApp.mxml on the local host, in the Flex context, and within a subdirectory named /sos, Flash Player stores the *.sol file in the following location on Windows:c:/Documents and Settings/username/user_domain/Application Data/Macromedia/Flash Player/#localhost/flex/sos/MyApp.mxml.swf/objectName.sol