Verification of signed message with encryption
There is some confusion in your question, because a signature in a public key cryptosystem is (usually) not just a hash, but a hash of the message that is signed using the private key. E.g. in RSA it would be a hash value padded and raised to the private exponent.

There are two ways to have an authenticated encryption in a public key system: Should we sign-then-encrypt, or encrypt-then-sign? In encrypt-then-sign, the receiver can verify the signature just as if they had received a signed plaintext message. In sign-then-encrypt they would first have to decrypt the message, but then they have just a signed plaintext.

In either case the message hash can be calculated by anyone since it does not include keys. The keys only come into play when verifying that the signature was taken over the hash, and there the public key is enough.

For low level details of how it works in the case of RSA you can refer to PKCS #1.