If someone suffered from multiple personality disorder, would they know?

Depends: Firtsly, dissociative identity disorder is very rare, and most of the "diagnosed" cases are severe forms of personality disorders. Regarding the awareness, the people who truly suffer from it may be unaware of the alternate personalities, but they will be aware of missing time or other clues. Sometimes, the individual will be aware of the others.

Multiple personality: In the early stages of treatment, likely the dissociated aspects of the personality would not be aware of much of anything wrong, but as the personality reintegrates, memory and therefore identity and awareness improves, up to and including knowing that at the very least, something is emotionally wrong with them.