You seem to be a bit confused about how asymmetric encryption works. There is no need for private keys to be transmitted for a messaging service to work. I encourage you to read about the difference between public and private keys in asymmetric encryption. They are generated in pairs, such that when something is encrypted using a public key, it can only be decrypted using the corresponding private key. So it's not correct to say that the message can't be decrypted by the intended recipient - they are in fact the only party who can, but even the sender can not.