I have found that continuing to hope in people who are showing no signs of stirring and no interest towards Christ is one of the most agonizing and mentally scarring experiences.
Perhaps that person will choose to ultimately reject God. Then what were all of my prayers for? I could have put that effort into prayers that could have been answered instead.
Your prayers, if sincerely prayed, were not for naught even if the persons never responded as God would like everyone to respond...yet you are not the first to be sent by God on such a mission:
"Therefore thou shalt speak all these words unto them; but they will not hearken to thee: thou shalt also call unto them; but they will not answer thee." Jerem 7:27
God sent Jeremiah to speak to the people, to tell them the truth, but He also told Jeremiah before hand that the people would not listen. Jeremiah was obedient and did as God told him and the people responded negatively also as God told him.
On the other hand, we see God calling Jonah to go and preach onto the wicked gentiles of Nineveh. Jonah ran from God but eventually he turned around and obeyed God. The Ninevites repented before God at the words of Jonah.
Why is it God bothered to send Jeremiah when He knew that there would be no success there with the children of Jacob as there was with the people of Nineveh?
Our prayers and words do not save people, but... they do cause hearts/doors to be opened so that people hear the truth. If they then reject the truth, the prayer warrior or prophet or evangelist or pastor has done his job but the judgment on the people is due to making an informed decision to disobey God.
"So thou, O son of man, I have set thee a watchman unto the house of Israel; therefore thou shalt hear the word at my mouth, and warn them from me.
When I say unto the wicked, O wicked man, thou shalt surely die; if thou dost not speak to warn the wicked from his way, that wicked man shall die in his iniquity; but his blood will I require at thine hand.
Nevertheless, if thou warn the wicked of his way to turn from it; if he do not turn from his way, he shall die in his iniquity; but thou hast delivered thy soul." Ezek 33:7-9
We, indeed, are God's watchmen or messengers or preachers, etc. but the test here is not only for those who we would like to see changed but also for ourselves as obedient servants/ministers.