[jrpt]

No, multiple parties could execute a smart contract between themselves, and keep the code private. Smart contract just means it's a contract that gets executed through code.

Even in the case of a public and decentralized platform like Ethereum, I don't think there's any requirement to make it open source. Further, I think with Ethereum, a smart contract is made up of compiled code and a Application Binary Interface (defines what functions can be called in your contract), so it could be possible to never even share the source publicly. I've never written an Ethereum smart contract so this is just my understanding, and I may be wrong.