TeamSpeak 3 Client is absolutely FREE to download and install on your PC, Mac or Linux Box. Once installed, you can freely access 1000s of public TeamSpeak servers or even your own private TeamSpeak server.
If you have your own server hardware, you can set up your own private server, completely FREE-of-charge too!
A license charge comes with bigger slot requirements or monthly costs if you rent a server for to be hosted by teamspeak or someone else as a service.
WHAT MAKES MYTEAMSPEAK THE NUMBER 1 CHOICE FOR PRO GAMERS?
Things listed there work.
What not works is (but is generally not listed as a feature):
Peer To Server Streaming (Only P2P possible as of now)
Text Only channels
Moderation of chats on TS6 servers (works in group chats)
See 2. It works P2P, so it is possible. However check if your ISP or settings allow P2P connections.