I've figured it out. The purpose here would be to encrypt data with your private key so that only people with your public key can decrypt it. I'm implying here that the public key is somewhat a secret between a single person or group of people and the owner of the corresponding private key, but this is not how public key cryptography works; theoretically everyone can have a copy of the public key. This implies that the public key is not a shared secret but public, and therefore there would be no point in encrypting data that anyone and everyone can see; it might as well be unencrypted!
To achieve what I am trying to do we use not asymmetric encryption like public key cryptography, but symmetric cryptography, commonly known as private key cryptography. In symmetric cryptography we encrypt data with a shared secret (a password for example, though any data can be used as the shared secret or passphrase), and this secret is shared amongst a group of individuals. As long as the shared secret is kept secret by only those with said shared secret then only those individuals can encrypt and decrypt data with that shared secret. This is what I was searching for. Unfortunately symmetric encryption does not provide identification and non-repudiation; all you know about the sender of symmetric key encrypted data is that it came from someone whom has the symmetric key, hopefully from a party that rightfully should have the key, that the key is still secret.
If you really want all the qualities of asymmetric cryptography (identification, confidentiality, integrity, non-repudiation) then get a group together and grab each others' public keys and start sending individual messages using each others' public keys. This doesn't work in the way you send bulk messages to multiple recipients like with private key (symmetric) cryptography, but that doesn't really matter when public key cryptography provides many more benefits, and really, is sending multiple messages really that hard?