One of the structures English uses for reciprocals is "one another". This, I've always interpreted as a sort of parallel implicit clause that tells us how to interpret the first clause. This is a very naive model, I know, but think of it as
(each) one verbs another
Russian has a rather similar structure, with друг друг- drug drug(obj/obl case), i.e. other(nom) other(object or oblique case), or even potentially friend friend(object or oblique case). In the case of Russian, I am pretty sure this is understood in terms of other other rather than friend friend – friend and other are homonyms but distinct lexemes despite sharing their etymology, but one could imagine a language where similar structures could be made from a much larger set of nouns or pronouns, possibly also including additional information, e.g. "child child(...)" would tell us that the group who acted reciprocally consisted entirely of children, etc. Of course, not all nouns would fit in there, and maybe there'd be some particular quirks, such as some slight reduction of the case markers or of the root or whatever, so e.g. "child chi's" for "each other's (wrt children)". One could go further with this idea, and for instance mark differences in the reciprocality - does each agent act on one other agent or on multiple ones, etc. (Not that that can't be marked by other strategies as well, and not that that really is all that interesting in most contexts.)