Round here (ZA) its fairly common to see a party having a copy of the contract signed only by the counterparty. i.e. Each party ends up with a contract signed by the counter party. Technically that makes neither copy a valid contract, but - close enough I guess.
Here too. And has its sense. You have the proof the other party agreed the contract. I do not need to sign to demostrate I am too. My word right now is sufficient.
This reminds me of the diplomatic practices of ancient mesopotamia -- when an agreement between two city-kings was finalized, each king got a master tablet listing half the agreement -- all of his own obligations. At no point did a symmetrical tablet listing every clause of the agreement ever exist.