Well, since it doesn't need to be in someone else's logbook, you don't really "need" to.

But it sounds like you might still have a paper certificate since when the FAA started issuing plastic they stopped using SSN's.

If that's the case, you might not be aware that all paper pilot certificates must be replaced. Fortunately, if the request for a new certificate includes a request to get rid of your SSN, there's no charge.