Personal is stronger and makes clear that she is your secretary and no one else's. Further, a personal secretary deals with more confidential issues, in most cases. Own implies that she is your secretary, but not as forcefully and doesn't carry the same level of meaning about confidentiality and privacy.

"She is my personal secretary." says what kind of secretary she is. She is a personal secretary, not a an executive secretary, nor a department secretary. "my own secretary" shows whose secretary she is. She is my secretary. She isn't your secretary or his secretary. She is my secretary. You can use both together, for example: "She is my own personal secretary." that shows whose secretary she is (yours) and what kind of secretary she is (a personal secretary).