The security of the AES secret itself
If a user has a copy of both the encrypted and decrypted data, he is in a position to perform at least a known-plaintext attack. If users can submit arbitrary plaintexts for encryption, they can conduct a chosen-plaintext attack, which is stronger. In a chosen-plaintext attack, the attacker can submit any number of plaintexts and can retrieve the corresponding ciphertext.

All reasonable encryption algorithms are resistant to chosen-plaintext attacks, and AES is considered a reasonable encryption algorithm. Someone in possession of some (plaintext, ciphertext) pairs cannot encrypt or decrypt other messages (except sometimes messages derived from the known messages if a bad mode is used). In particular, no matter how many known plaintexts and ciphertexts the attacker is, he has no method to obtain the key that is better than brute force (trying all possible $2^{128}$ keys).

Do note that deleting the private secret does not delete the master secret. The master secret can still be recovered from backup copies of the private secret that weren't deleted, from other users' private secret, or from extant copies of the master secret.