The minhag at kiddush levana is to wish shalom aleichem 3x. It is prevalent to do so even if someone was wished shalom aleichem and he reciprocated back aleichem shalom. Is there a reason that one needs to be the initiator of the blessing?
Because we just just finished saying a few verses with curses for our enemies, we then say Shalom Aleichem at Kiddush Levana to those around us, to highlight that we were not referring to them.
The Shulchan Aruch, Orach Chaim 426:2 (page 574 of vol. 4 of the Mishna Berura) quoting the Tur, says that "answering is like asking", and implies that you would not have to say Shalom Aleichem if you already responded Aleichem Shalom to 3 people.