Skype says that the new call recording feature is cloud-based, with Skype informing every participant in a given call that a call is being recorded.
Call recording for video chats includes everyone's video and screens that are shared during the call, with the call available to be saved and shared for the next 30 days. There is no way to record the audio portion of a video chat without also recording the video.
On both mobile and desktop, call recording can be initiated by clicking or tapping on the "+" icon at the bottom of the screen and then selecting "Start recording." Once call recording has been initiated, participants will see a mobile banner letting them know that a recording is in progress.
Call recording is available in the current version of Skype, which can be downloaded from the Skype website on the Mac or the App Store on iOS. [Direct Link]