Note: This is related to Lighting the menora in shul and at home. How many brochos need be said?, except that in that case one of the occasions involved a minyan, and the explanation offered in the accepted answer there seems to depend on that fact ("the reasoning is that each new minyan is a new obligation, even if everyone in the minyan already lit individually"). Also related is Menorah lighting at work and saying shechechiyanu, but that question is currently on hold.
Yesterday evening, the first night of Chanukah, my family visited my mother's home, and we all (myself, my wife, and our children) lit the chanukiyah at her house. To be more precise, my mother lit the candles, but we all sang the b'rachot in unison together.
Later that evening when we returned home we lit our own menorah. Naturally the question arose: Should we recite shehecheyanu? What about the other brachot?
On reflection, it seems a more fundamental question is: Is it appropriate to recite b'rachot communally, when only one person is actually performing the mitzvah? This is how my family has done it since I was a little kid, but that of course doesn't mean it's halachically proper.