Do people with multiple personality disorder know that they are suffering from it?

In some ways: The central or personality or "alter" may be aware of the others, but some of the others may be unaware of the rest. All parts probably experience some suffering, but may not perceive the problem as one of "multiple personality" [actually now known as dissociative identity disorder]. Therefore, the answer to your question depends on which alter is in charge when you speak with the person.