NEC3 ECC: Do both parties have to sign an ECS contract for it to be valid

Practically speaking yes, but contractually speaking not necessarily. Pretty sure the lawyers will confirm that a contract exists if the Parties are proceeding with the works even though the contract itself has not been signed. It would be the contract that the parties tendered upon - which can cause all sorts of issues if the reason that the contract was not signed by one Party is that they did not agree with some of the terms - but in the meantime are underway with the works!