Zend certified PHP/Magento developer

Can the public key be extracted from the message encrypted with it?

If I encrypt messages with Public Key, and give the recepient Private Key to decrypt it, can the recepient somehow extract the Public Key from the messages?

I.e. is it OK to keep “Public Key” in secret, to encrypt messages, and give the “Private Key” to the third parties so they could decrypt messages, but not create their own with the same Public Key?