Can any Bitcoin client do local peer discovery and blockchain synchronization? Usually, when a full Bitcoin client such as bitcoind, Bitcoin-QT, or Multibit start on a fresh installation, they connect "to the network" to download the blockchain. If someone has multiple computers or devices, they could end up downloading the whole blockchain several times. This puts a lot of stress on their connection and extra clients on the network that could get their data locally.

Have any clients implemented a standardized way to exchange blockchain data locally, without going over the Internet?

I imagine this would work something like how BitTorrent local peer discovery works, where two clients on the same subnet discover each other through multicast groups and then negotiate transfer of data.
The Bitcoin.org client can be launched with the -connect= parameter.

Simply have it connect to another local node and no outside connectivity occurs.

But once it is caught up to the latest block, the amount of bandwidth is relatively trivial.  There's no reason to do this to save bandwidth once a node has been brought current.