Is there a way of rewarding users who confirm others' transactions? Users who confirm or validate others' transactions are performing a public service on the Bitcoin network.

Is there any way to reward such users, perhaps by giving them priority for confirmation of their own transactions?
This is the purpose of transaction fees. By mining a block, all transactions hashed in the block are confirmed, and the corresponding transaction fees go to the block discoverer (along with the block reward).
Right now the block reward dwarfs the transaction fee reward... but there is no downside to including other transactions while mining, so all of the nodes will do it.