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Recently, non-reciprocal systems have become a focus of growing interest. Examples occur in soft and active matter, but also in engineered quantum materials and neural (brain) networks. Here, we investigate the impact of non-reciprocity on the collective behavior of a system of (dry) chiral active matter. Specifically, we consider a mixture of "circle swimmers" with steric interactions and non-reciprocal alignment couplings. Based on hydrodynamic equations which we derive from a set of Langevin equations, we explore the interplay of non-reciprocity, finite size, and chirality. We first consider, as a reference, one-species systems with reciprocal couplings. Based on a linear stability analysis and numerical simulations, we here observe three different types of collective behavior, that is, flocking, motility-induced phase separation, and a combination of both. Turning then to a non-reciprocal system, we find that non-reciprocity can turn otherwise stationary instabilities into oscillatory ones, affect the relative orientation of flocks, and, crucially, change the general type of instability. This illustrates the drastic impact of non-reciprocity and chirality on the emergent collective dynamics of active matter systems, with potentially far-reaching biological implications.