For example, imagine that you use Google Groups for general discussion, Google Mail for direct communication and Google Documents for specific papers. All your data is on Google's servers, you have no locally saved copies (usually.) Now Google is offline, or your account is disabled (all of that happened several times already.) You are in trouble, isolated from your contacts and robbed of all your documents and messages.
An opposite example is FreeNet. This is an experimental network that is designed for storage of documents on multiple computers (like p2p.) No single computer holds everything, and nobody knows where the data is actually stored - the network is designed to ferret the data out, but it won't tell you where it found it. So an attacker has no easy way to disrupt the network.
Travis, this sounds like the kind of network some of your characters might need sometime. Analogous to what you were looking for for phones. This is something that would take a few people to set up and would only be useful if quite a few were using it, so it would only be a feasible story element in a scenario when a substantial number of people were leery of the government to that degree. But your books all have that covered. ;-)
In my first novel (2003) a computer network guy posted the names and particulars of all of the federal agents in several states on a dispersed system somewhat analogous to gnutella and other music programs. I’m not sure what’s possible today.