It's likely that anyone who wanted to operate a major server would be tied in to the network by one or two 1+gbps links, and probably colocating with other servers, allowing them to share the overhead of this, and have several connections operating to the network in parallel, lowering the stress on any one route.
Similarly, a lot of traffic is aggressively cached now to mitigate the need to hit the server with an identical request. There's no reason to think that certain kinds of content wouldn't be cached at various points in the mesh, making the total traffic to any particular server lower still.
A realistic scenario for a datacenter is that 50-100 people split the cost of having 10 1+gbps links to the network, hitting it at multiple points well spread in the network. The inside of the data center would do some routing to handle the load balancing. I expect the total cost of that would be about $35k to set up, but splitting it between that many people, each person is paying (on average, for 50 to 100 people) between $350 and $700 - well within a reasonable amount for a coop to set up a datacenter. Further, $35k is a small enough price tag that it would be acceptable for a business (or a couple businesses) to share such a datacenter themselves, if there were a particular incentive for them to operate as part of a community mesh, rather than with the current internet.
[1] http://www.amazon.com/Ubiquiti-AF24-Airfiber-Wireless-Comple...
[2] http://www.amazon.com/Ubiquiti-AF-5-airFiber-5GHz-Wireless/d...
I guess it would be useful to gauge the size of local vs international traffic - Facebook does not need to send my data to SF unless I am in contact with someone in SF. Video uploads and downloads are similar. Most of my daily traffic almost certainly has a strong correlation to my physical location.
If it just so happened that you are one of the few users with a mesh-network connection to a peer across the street, much or all of the traffic between your street and the next street will pass through you. Since it's a mesh network, you are expected to have a low-power, low-throughput node, which might be overwhelmed.
Do not conflate mesh networks (of nodes in physical proximity) with peer-to-peer networks (of nodes in logical proximity via all of the Internet).
There is no particular reason a mesh network needs to be low power and low throughout - I think one of the most potent mesh network opportunities is (slowly?) replacing in home routers with (firmware?) to communicate amongst a neighbourhood. With that a small and growing group of people will be able to exchange local data and video and voice.
I perhaps naively think that physical proximity will become a major deciding point in bandwidth between two nodes (which makes sense) and this will encourage an explosion of useful local perhaps democratic applications - think of it as garden fence protocol. Sharing not just pirated films but kids calling their neighbours to just chat, parents arranging bar eques etc
Billions have been spent to create the IP transport networks. Building and operating them at current capacities, or even at 1% of current capacity, is of out of reach for most people who would run peer nodes of a mesh network.
Any node that'd happen to be in the bottleneck connecting two large segments, or just on a route to a very desired host, would be hit by a huge demand to pass traffic. Top-of-the-line 300 mbit wi-fi will be easily choked by that, not giving the node nearly enough bandwidth for its own requests. Huge fiber cables and microwave lines exist for a reason, and are not going to disappear.
The shape of the traffic they carry might change, though. E.g. with the advent of bittorrent, much of the traffic worldwide has become peer-to-peer / decentralized. This did not affect major carriers too much, except that they had to add bandwidth.