On the global web things are a bit different then in a LAN situation in which it would be fairly easy.
The point is that you have little control over the routing, if one could manipulate one such route there is a opportunity there.
Don't underestimate how hard this can be.
As far LAN there are several ways of doing such a thing.
You could set the same IP address on your box, as for knowing the required address you can simply bruteforce that.
Then you would have to kick the other machine out of the network , one way to do it is to bombard the host with ARP packets so it will use your MAC in its cache