And as one would use the money, the other one will see the transaction and know it can't use it anymore
this is why you can use the same Electrum wallet on different devices.
Two Electrum wallets that have the same seed will use the same sequence of addresses, and remain synchronized on different machines, with no need to communicate.
Interesting... I tried Electrum wallet, but I could not figure out how to import a private key... Tried several unencrypted formats (base 58, base 64, HEX, Bitcoin-qt), always the same error: The following inputs could not be imported...
UPDATE: I was using the wrong password. I could import the private key with the Bitcoin-QT format...