In this paper we examine interactions of the reciprocal with distributive and collective operators, which are encoded by prefixes on verbs expressing the reciprocal relation: namely, the Czech distributive po and the collectivizing na-. The theoretical import of this study is two-fold. First, it contributes to our knowledge of how word-internal operators interact with phrasal syntax/semantics. Second, the prefixes po and na generate (a range of) readings of reciprocal sentences for which the Strongest Meaning Hypothesis (SMH) proposed by Dalrymple et al. (1998) does not make the right predictions. The distributive prefix po prefers the Strong Reciprocity reading, although the SMH predicts that a weakening should take place, while with the prefix na we find cases where weaker reciprocal readings are preferable to the stronger ones predicted by the SMH. This behavior of po and na is, we propose, due to the way in which they modulate two factors that are crucial in the interpretation of reciprocal sentences: (i) the relevant subpluralities in the group denoted by the reciprocal's antecedent, and (ii) the strength of reciprocal relations. We provide a detailed analysis of the semantics of the prefixes po and na and their contribution to the meaning of reciprocal sentences within the general framework of event semantics with lattice structures.