The sneezer himself
"[Talmud Yoma 88a: the breath of life in his nostrils teaches us that the essence of life is in the nostrils.] That is to say, the essence of the distinction by which one knows if the person who seems dead is definitely dead, or if there is still some breath of life - if there is still some breath in his nose there is still the breath of life, and if not he is definitely dead. One should not rely on other definitions. The reason for this, it seems, is that a person's soul leaves him in the way which it came, and since it came first through the nose, as it says in his nostrils the breath of life, so too is its exit also through the nose.
"It's possible that this is the reason for the custom to say "Asuse!"[Aramaic: Health!] to the person who sneezes (see Berakhos 53a and Rashi there). This according to what is said in midrash Yalkut, parashah Lech lecho, which indicates that until Yaakov people did not become sick before death but died suddenly by sneezing - a person would sneeze and his soul leave through his nose. This is also because the nose is the transit point between life and death, therefore when a person sneezes he is exposed to danger, and people say to him "Asuse!" Rashi writes in Berakhos: 'People are accustomed to say Asuse to the person who sneezes.' That is, only other people customarily say this to the sneezer and not the sneezer himself. This also implies that this is but a custom. But the Yalkut there maintains that a person is obligated upon sneezing to thank God - apparently then the sneezer himself is required to say some words of thanks. There are those whose custom it is to say 'I hope for your help, God' [lishuoskho kivisi adoynoy]. Apparently this is then a obligatory custom. Possibly one can say that the sneezer himself is obligated, but for others it's only a custom out of politeness [derekh erets]. But I'm not going to go into this further."
--Borekh Halevi Epstein, Torah Temimah on the Torah portion Noach