This idea involves combining several technologies, all of which individually already exist. That doesn't mean that it's already baked.
The problem : Personal computers just arn't fast enough. They're fine for most purposes, but occasionally you need a bit more out of them and they have to think for
a minute or two. This isn't acceptable. At the moment most programs use a single processor core, but as computers gain more cores as standard this is likely to change.
Background information : The internet, as it appears to home computers, is not a particularly localised entity. I think the typical case is an internet connection but no home network. In this case your computer can talk to your router, which can talk to your service providers computers, and after a few hops pretty much any other accessible computer.
This means overall that there are an awful lot of computers with a single connection, and relatively few which have many connections.
Now, I said your computer is fine for most purposes, but that's not quite true. Most of the time it's more than fast enough - it's sitting around twiddling its thumbs and wasting power.
My proposal: Computers are interconnected with nearby machines in some standard way. As the nearby machines probably are your neighbours, this probably involves agreeing with them to drill little holes in the walls and feeding cables though. Wireless communications are fine too, if they're expedient. Okay, hook everything up with a suitable router. Now there should be very fast communication between neighbouring machines - probably faster even than internet access, because your service provider is further away both in terms of distance and number of 'router hops'. Also it's possible that a simpler communication protocol could be used.
Now, this system needs some software and perhaps operating system support to make it work. This would probably work best with a system built from scratch to support it. Data from neighbouring machines needs to be sandboxed, and processing would have a lower priority than your own stuff.
Now if all that works, when your computer experiences a spike in processor load, it can farm that out to your neighbours machines, provided that they're idling. Of course, your machine reciprocates the service. There should be a process in place to make sure that usage is fair - if anyone tries to abuse the service by not honouring any processing requests they can be easily cut off.
Potentially for big jobs your neighbours could pass some things along to their other neigbours for even more speed. Latency would be great for local access but wouldn't scale well with distance. However, when your main internet connection breaks you'd still be able to talk to the world via some guy with another service-provider. Even in normal usage it may increase your burst download rate a bit.
Regarding privacy, there would probably need to be a 'work in private' option for when you're working with confidential information. Perhaps files could be tagged 'private' and never exported, too.
Once a system like this is in place, your computer need never be wasted again; you could sell (or donate) unused cycles to people with enormous massively parallel jobs. These would run with even lower priority so as to not displace your neigbours jobs. This is a big improvement on the current practice of when your screensaver is active, because your computer is probably idling a lot between keystrokes.
Oh, why that title? It's a local cloud.