No liability for how third parties using your network (e.g. for libel or violating copyrights). There may be other regulations that limit or eliminate this liability though.
So, right now, if someone detonates a bomb with a cell phone, Verizon/AT&T is basically a co-conspirator? You'd think they'd want to be a common carrier, at least if it weren't for the US's general lack of enthusiasm for holding corporations responsible for anything.
Um, no. That common carriers aren't liable for something doesn't imply non-common carriers are liable. Moreover, IANAL, but I can't imagine common carriers are exempt from prosecution for criminal conspiracy.