Is it possible to record video through Flash's "Webcam" facility and receive encoded video on server side without any further processing (i.e. just saving the stream to a file)? Or is server side software always needed to encode the input?
If server side software is necessary, are there free / Open Source solutions available?
Yeah, you have to use server side software to capture and distribute. There are open source solutions: http://osflash.org/red5. I tried using this and it was too much of a hassle to install, though, so I just used Flash Media Server http://www.adobe.com/products/flashmediaserver/ hosted by another company.
You can also use the Nimbb API, which will allows you to record videos and download the FLV file by calling a web service. You don't need any server side setup.