The exchange or the receiver does not need to know if a second signer is needed because the transaction will not be broadcasted at all. Only when the transaction would be broadcasted to the mempool is when the second signer sign the transaction.
Sorry, still not getting it. In the "2 of 4" scenario, let's say you signed a transaction on your wallet (eg, Ledger). The next day I sign the transaction using my wallet (eg, Trezor). Does my wallet somehow know you already have signed, and thus can broadcast the transaction? If yes, how?