What is the impact of purposely sending a signed message with zero fee? Would it be appropriate use of me to send a message with 0 fee on purpose if I have no need for the message to remain on the blockchain after it is seen by the recipient?

I could sign a transaction with 0 fee and provide the recipient with the txid. It would not hurt me if the transaction was confirmed but it would also provide me with no benefit once the (pending transaction) is seen by the other party.

Would I be helping the network by providing no permanent bloat of no value (assuming my txid is never confirmed) more than I am hurting it by adding no miners fee?
The underlying motivation behind transaction fees is to compensate for resource utilisation as there is always an inherent cost when validating transactions (Electricity - CPU usage) and relaying transactions (Internet Bandwidth). Another reason why there's transaction fees is also to deter spam. 

By purposefully sending a signed message (with no fees) to the Bitcoin network, what you're effectively trying to do is use these scarce resources for free and therefore it is perfectly possible for participating nodes to refuse to relay/process your transaction.

However, you may use a signed message with zero fees to prove that you're in control of a unspent transaction output for example. Instead of broadcasting this directly to the Bitcoin network, what you might do instead is send this to one party who requires this "proof" privately and so as long as the UTXO is valid (i.e. committed in the Blockchain already), then your signed message with zero fee is as convincing as it gets to sufficiently prove that you have the funds to do whatever it is you want to do.