It is possible to go serverless with Flash. This is doable with Adobe Flash's Peer to Peer capabilities. I once wrote a peer to peer chat with it. The drawback is Actionscript is a dying language and may not be supported much in the future.
Here is the raw class.
http://help.adobe.com/en_US/FlashPlatform/reference/actionscript/3/flash/net/NetGroup.html
Here is resources if you don't want to write your own.
http://www.as3gamegears.com/category/multiplayer/
If you want a Server option that is light on the server side. Try this node.js extension.
http://socket.io/
I recommend using a java socket server of some sort. Electroserver used to be one of the leaders in the field, it had Unity support and was scalable to hundreds of thousands. Although I think they have fallen on hard times. The Electroserver site has not been accessible for sometime. I know there are others out there but Electroserver is the only one I have used.