If someone is being yotzei with another’s bracha, do his hands have to be clean in the same manner as the one making the bracha?
Hands must be clean in order to answer amen to the berachah.
However, this is because answering Amen is considered a matter of Kedusah. For actually being yotzei, there is no need to hands to be clean.
Concerning saying Amen with dirty hands, see Shut Shevet HaLevi (7:5), and see also Halachah Berurah (1, Otzros Yosef 16), who concur that this is considered a davar shebikdusha and requires washing.
Yet, this refers to saying Amen, and not to actually being yotzei. For the matter of being yotzei, the question might depend on whether or not it is considered as though the person hearing is actually saying the berachah, or not (many explain the dispute between the Beis HaLevi and the Chazon Ish concerning Birkas Kohanim along these lines).
Since the more simple understanding is that the person being yotzei is not “actually” saying the berachah, it follows that one would not have to wash for this purpose.