Social network alignment shows fundamental importance in a wide spectrum of applications. To the best of our knowledge, existing studies mainly focus on network alignment at the individual user level, requiring abundant common information between shared individual users. For the networks that cannot meet such requirements, social community structures actually provide complementary and critical information at a slightly coarse-grained level, alignment of which will provide additional information for user alignment. In turn, user alignment also reveals more clues for community alignment. Hence, in this paper, we introduce the problem of joint social network alignment, which aims to align users and communities across social networks simultaneously. Key challenges lie in that 1) how to learn the representations of both users and communities, and 2) how to make user alignment and community alignment benefit from each other. To address these challenges, we first elaborate on the characteristics of real-world networks with the notion of delta-hyperbolicity, and show the superiority of hyperbolic space for representing social networks. Then, we present a novel hyperbolic embedding approach for the joint social network alignment, referred to as PERFECT, in a unified optimization. Extensive experiments on real-world datasets show the superiority of PERFECT in both user alignment and community alignment.