It is the IV shared in CBC mode? I don't know if the IV in CBC mode is equal in both sides of the encryption and decryption operations. 

I know that the IV must be unpredictable, but has it to be MACed in the encryption side to be deMAC and "rebuilt" to use it in the decryption side ?

So you need to "transport" the IV along with the cypher-text to use it in the decryption ?
Yes, the IV is the same in both encryption and decryption for all modes that use an IV.  And yes, the decrypter must have access to the IV to decrypt the ciphertext.  The IV is not a secret.

The only caveat is that for CBC mode the IV must be unpredictable to an attacker whose data is being encrypted by another party.  Once the attacker has presented the data to be encrypted that IV need no longer be secret.