When it works, AirDrop between Apple devices seems like it does exactly what you describe. Of course, this assumes that the two devices can see each other via Bluetooth and I'm pretty sure the thing is partly mediated by iCloud.
I'm not sure if there's an equivalent in the Windows world.
> ...establish a direct connection between two web browsers and directly push files through
Is this really a web browser's job? Why does it feel like the default answer to every "Why can't I do X easily?" question is to make the browser do it?
I used to transfer stuff between different Windows machines (on the same network) by simply going Start, Run, \\OtherComputerName and authenticating. Regular users, like my parents, would just go to Network Neighbourhood and find the computer in question. It was pretty easy and all built in to the OS.