The framing of political issues can influence policy and public opinion. Even though the public plays a key role in creating and spreading frames, little is known about how ordinary people on social media frame political issues. By creating a new dataset of immigration-related tweets labeled for multiple framing typologies from political communication theory, we develop supervised models to detect frames. We demonstrate how users' ideology and region impact framing choices, and how a message's framing influences audience responses. We find that the more commonly-used issue-generic frames obscure important ideological and regional patterns that are only revealed by immigration-specific frames. Furthermore, frames oriented towards human interests, culture, and politics are associated with higher user engagement. This large-scale analysis of a complex social and linguistic phenomenon contributes to both NLP and social science research.