of course a local area network (lan), which is what you seem to be describing in your "peer-to-peer," doesn't require external infrastructure other than the electricity it uses to run its local equipment and the massive technical and transportation infrastructure required to produce that equipment.
but it also does not allow communication outside that local network, which is the primary value (in this context) of "the internet". the internet is what allows multitudes of those local networks (and the individual people/computers on them) to communicate with each other, opening up the possibilities that are so highly touted (and from some perspectives, rightly so).
while i don't have (or particularly need for my own understanding) statistics, some of the major sources of power involved in running a global network include: obtaining all the resources and requisite extraction to create/build the actual technical infrastructure (computers, networking hardware, cables, monitoring equipment, massive cooling systems, buildings to house them, security to protect them, people to program them, ...); as well as everything needed to actually run it reliably and perpetually.
i realize that is extremely high level and non-specific, but i don't have time right now to go into greater detail.
the actual performance (as in speed of communication) of the internet is not what i am referring to here. shit, we could all use sneaker-net to communicate, especially in your local area network. just not so useful if you want to communicate with someone on another continent, say.
also (to bring it back to the original question), everything i mentioned as being required to create and maintain an internet infrastructure requires massive specialization, something that i personally consider to be largely antithetical to my anarchy.
it is impossible for me to imagine a "sustainable" internet. and though i use the internet frequently and for many purposes, i would love to see it gone, along with most of the rest of what human technological "progress" has wrought (and even worse, what it threatens to bring about, eg the singularity, brain/computer melding, etc).