I am wondering if there exists some asymmetric encryption algorithm that would work with two different private keys, so that the encrypted message can be decrypted by either one of such private keys?
Asymmetric keys are never used for encrypting messages (too slow, has a size limit, etc.), only for encrypting small buffers, like symmetric keys for messages.
When you encrypt a message with an asymm. key, you are in fact encrypting a symmetric key which encrypts the message.
Simply encrypt the same symmetric key with two different asymm. keys and you'd get two different cipherblocks which both contain the same symmetric key for the message, so you can decrypt with either one.