Recent years have seen numerous attempts to imbue conversational agents with marked identities by crafting their personalities. However, the question remains as to how such personalities can be systematically designed. To address this problem, this paper proposes a conceptual framework for designing and communicating agent personalities. We conducted two design workshops with 12 designers, discovering three dimensions of an agent personality and three channels to express it. The study results revealed that an agent personality can be crafted by designing common traits shared within a service domain, distinctive traits added for a unique identity, and neutral traits left intentionally undecided or user-driven. Also, such a personality can be expressed through how an agent performs services, what contents it provides, and how it speaks and appears to be. Our results suggest a renewed view of the dimensions of conversational agent personalities.