From inside your Flash asset, you'll need to create buttons that invoke sharing to a given service such as Facebook and Twitter. Link these buttons to our sharing endpoints for each particular destination (described here). For example, the following link can share a YouTube player to Facebook:
http://api.addthis.com/oexchange/0.8/forward/facebook/offer?url=http://www.addthis.com &title=Addthis%20Video &swfurl=http://www.youtube.com/v/1F7DKyFt5pY&hl=en&fs=1 &screenshot=http://i2.ytimg.com/vi/1F7DKyFt5pY/default.jpg &height=560 &width=340
We highly recommend pointing your URL to an HTML page containing meta tags that describe the Flash content, instead of just specifying it on the call; this is also a best practice for sites like Facebook, search engines, and more.
If you don't want to write the Flash code to do all of this, take a look at our Flash Example Code. We offer both a low-level sharing API as well as a full Flash-native sharing menu.