While social media has become an empowering agent to individual voices and freedom of expression, it also facilitates anti-social behaviors including online harassment, cyberbullying, and hate speech. In this paper, we present the first comparative study of hate speech instigators and target users on Twitter. Through a multi-step classification process, we curate a comprehensive hate speech dataset capturing various types of hate. We study the distinctive characteristics of hate instigators and targets in terms of their profile self-presentation, activities, and online visibility. We find that hate instigators target more popular and high profile Twitter users, and that participating in hate speech can result in greater online visibility. We conduct a personality analysis of hate instigators and targets and show that both groups have eccentric personality facets that differ from the general Twitter population. Our results advance the state of the art of understanding online hate speech engagement.