Short answer: Yes. Long answer: Not really. Messages in Words With Friends are handled by the game's developer, Zynga. Zynga has been historically ethical with this type of personal data. That said, anything you send inevitably goes through their servers so it can be read by your opponent. It stands to reason that this data may linger, unused, for some time after games are completed for legal and moderation reasons.
Your opponent cannot 'trace' your messages. The developer, and the law by subpoena, may still have enough information to track users for their purposes agreed to in the terms of service.