Good question. We use CDIs in the mental health field for cases when the terp is not sufficent for determining the cultural influence on communication. Our CDIs do more than interpret from one language to another; they actually interpret context from a cultural perspective. That is very important when doing an assessment because it prevents bias and misdiagnosis when answers to questions are interpreted from a hearing perspective.
If a formally oral deaf person is skilled in interpreting from a contextual perspective, then I don't see why they can't function as a CDI. By the same token, all the CDIs I have ever had contact with are native signers.
You've given me something to think about.