Não, eles não pagam para entregar o Profeta Diário.
Hagrid explica por que a coruja queria pagamento - era para entregar o jornal. O jornal custa dinheiro, então a coruja que o entrega tem que recolher o dinheiro e devolvê-lo ao Profeta Diário. Não parece que os pagamentos são realmente para as corujas , mas que eles coletam o dinheiro para o Profeta Diário e entregam o dinheiro a eles da mesma forma como entregam as mensagens.
“The owl swooped in and dropped the newspaper on top of Hagrid, who didn’t wake up. The owl then fluttered on to the floor and began to attack Hagrid’s coat.
‘Don’t do that.’
Harry tried to wave the owl out of the way, but it snapped its beak fiercely at him and carried on savaging the coat.
‘Hagrid!’ said Harry loudly. ‘There’s an owl –’
‘Pay him,’ Hagrid grunted into the sofa.
‘What?’
‘He wants payin’ fer deliverin’ the paper. Look in the pockets.”
- Harry Potter and the Philosopher's Stone, Chapter 5 (Diagon Alley)
Os corujas recebem mensagens gratuitamente, mas parece que há um serviço de assinatura em que os assistentes podem ter o Profeta Diário entregue a eles por coruja e pagar quando a coruja chega com o papel. Hermione também recebe o Profeta Diário e tem que pagar a coruja que o trouxe.
“Hermione, however, had to move her orange juice aside quickly to make way for a large damp barn owl bearing a sodden Daily Prophet in its beak.
‘What are you still getting that for?’ said Harry irritably, thinking of Seamus as Hermione placed a Knut in the leather pouch on the owl’s leg and it took off again.”
- Harry Potter and the Order of the Phoenix, Chapter 12 (Professor Umbridge)
Outra vez Hermione é entregue o Profeta Diário, ela tem que dar o pagamento para a coruja também.
“Hermione was not the only person eagerly awaiting her Daily Prophet: nearly everyone was eager for more news about the escaped Death Eaters, who, despite many reported sightings, had still not been caught. She gave the delivery owl a Knut and unfolded the newspaper eagerly while Harry helped himself to orange juice; as he had only received one note during the entire year, he was sure, when the first owl landed with a thud in front of him, that it had made a mistake.”
- Harry Potter and the Order of the Phoenix, Chapter 26 (Seen and Unforeseen)
As corujas do Profeta Diário presumivelmente retornam ao Profeta Diário com o dinheiro que recebem. O dinheiro parece ser pagamento para comprar o jornal em si, não para a coruja que o entrega.