I think it's quite likely that a second internet would indeed be created, much like Usenet was originally made in response to the original Internet being academic and military only.
In a way, it actually is pretty cool, as the younger generations are increasing more connected to each other, and less to existing orders (see Arab Spring, OWS, etc.)
Whether this is a good thing is debatable. The resulting citizen network would unquestionably be freer, but there's serious question as to its reliability. Usenet, though awesome, relies on an ad hoc network infrastructure where it takes a few hours for most people to see posts (and very occasionally a couple of days if posting from a poorly connected server with issues). This works fine for forums and file sharing (which are the 2 things it's for), but setting up that kind of network to serve chats, games, etc, would be much more difficult. Of course, one could also argue the only reason it hasn't been done is because there hasn't been a reason to, and this would change it.
The other possibility, unfortunately, is that paranoia may breed a series of shadow networks, many of which are highly insular, and it could be the end of the whole "one network does everything" ideal that we run on now.
In short - it adds chaos to the system. *If* the right people channel that chaos positively, the result could actually be quite awesome.
