You want to create a Voice over IP (Internet)....
This cannot be done without Internet...
You can still achieve Voice / Data within your Local Network by dialing the Extensions.. but without internet you WONT be able to access your Extensions outside your network or other Phone numbers.
And with the Browser API... its already created.. The Technology is called SIP (Session Initiation Protocol) and its the technology that pushes VOIP.
You can get SIP for:
Web-browsers (or get the code to run on your own server)... For PC/Linux, Mac/Unix
Android & IPhone apps... WebOS apps... all SmartPhone, ChomeOS environments.
Linphone is a popular one and free as well. http://www.linphone.org/technical-corner/linphone-web/overview
And you also have BIG NAMES... like x-Lite and 3Com.
Asterisks is a Switch Board ONLY... routing calls for you the way an operator would... but you still need service.. to go beyond your network... otherwise your just texting/chatting/video via LAN only.