Authentication Schemes

Two parties are trying to authenticate each other using a shared private key. Listed below are two possible schemes for authenticating.

Scheme 1:

1. A -> Establish Connection -> B
2. A -> Send Challenge (X) -> B
3. A <- Send Response Encrypted(X) <- B
4. A validates encryption
   
5. A <- Send Challenge (Y) <- B
6. A -> Send Response Encrypted(Y) -> B
7. B validates encryption
8. Authenticated!
   

Scheme 2:

1. A -> Establish Connection -> B
2. A <- Send Challenge (X) -<- B
3. A -> Send Response Encrypted(X) -> B
4. B validates encryption
   
5. A -> Send Challenge (Y) -> B
6. A <- Send Response Encrypted(Y) <-B
7. A validates encryption
8. Authenticated!
   

Can either or both be exploited? How? Are there multiple vulnerabilities?