Adobe Flash Player 10.1 abides by the host Browser’s “Private Browsing” mode
Windows, Linux, Mac: Upcoming Adobe Flash Player 10.1, still a beta 2 pre-release, will introduce many new features like the Windows-only H.264 video hardware acceleration support. In this new version flash will accept without objection your wish to browse in the private mode, it will automatically clean-up your flash history once you terminate the session.
Flash Player 10.1 abclasses by the host browser’s “private browsing” mode, where local data and browsing activity are not persisted locally, providing a consistent private browsing mechanism for SWF and HTML content. Private local shared objects behave like their public variants as long as Flash Player is in memory and local shared objects created during private browsing are removed when returning to public browsing mode. Existing shared objects are preserved but inaccessible until private browsing is turned off. Libraries in the Flash Player cache, like the Flex framework, are unaffected by private mode. Supported in Firefox, Chrome, and Internet Explorer. No developer action required.
According to the release notes of Flash Player 10.1 beta 2 (extract above), this will work only on desktops and on web browsers with the “Private Browsing” mode; Firefox, Google Chrome, and Internet Explorer. Opera 10.5 pre-alpha now has private browsing but it might not be able to “force” flash into private browsing.