You can send data directly to a remote computer; however, this machine must be tapped, and if you are not using Air, this machine needs a socket policy file. Obviously, it is impossible to connect several machines directly to each other without a policy file, forces you to have a central server and prevents direct implementations in p2p chat / video / (w / e) applications in the browser.
So you should have a central server; however, you do not need to pay for them.
I knew I read something about this, so I searched google and came up with the links below.
http://haxe.org/doc/flash/peer2peer
https://github.com/OpenRTMFP/Cumulus
All you need is a developer key, which Adobe gives for free:
https://www.adobe.com/cfusion/entitlement/index.cfm?e=stratus
With that said, you need to know some C ++ to take advantage of this.
If you want to learn how to do something basic to get started, and you are really interested in creating something for your local network, then these articles describe how to make RTMFP as a multicast group.
http://www.flashrealtime.com/videotutorial-remote-device-controller/
http://www.flashrealtime.com/local-flash-peer-to-peer-communication-over-lan-without-cirrus/
[EDIT: Content for the last two links since the site deleted these pages can now be found using waybackmachine and snapshots in early 2011]
There are tons of things you can do with the information above; However, I would start small.