I didn't exactly mean monitor an entire wallet. My phrasing wasn't good.
I meant this:
One can transfer BTC to an address not on the network. Or even on the network. It would be nice if there were a chrome or firefox extension to monitor activity going to any given address (using Block Explorer).
You can transfer any bitcoin to any address you like whether or not the user that own the address is online or not. Wallet.dat is simply required for controlling the bitcoin in that specific address. It's more like a keychain than a wallet that hold your cash.