Can you decrypt all messages if a private key is obtained of one of the two entities in asymmetric encryption? Given a context where Alice sends messages to Bob and vice versa. There is an intruder, Carol, that has obtained the private key of Alice. She also obtained both public keys. 

Can Carol decrypt all messages or only the ones that are sent from Bob to Alice?

If Carol can decrypt all messages:
How can she do this?
Are there algorithms where Carol couldn't decrypt all messages if she obtained only one private key?
If they both use public key encryption to encrypt messages (think e-mail + GPG for example) than obtaining Alice's private key would allow you to decrypt only messages sent to her from Bob. To decrypt messages sent to Bob you need his private key. This of course assumes that there were no plain-text copies of messages in the same place where attacker obtained Alice's private key. 

In case they use some chat-like protocol (e2e encrypted messenger) and they use their private keys to set some session keys things are different. When Eve has Alice's private key she can read all messages sent in either direction after she obtained key and potentially all past messages (again, in either direction) if protocol used has no forward secrecy