Collectives gather online around challenges they face, but frequently fail to envision shared outcomes to act on together. Prior work has developed systems for improving collective ideation and design by exposing people to each others' ideas and encouraging them to intermix those ideas. However, organizational behavior research has demonstrated that intermixing ideas does not result in meaningful engagement with those ideas. In this paper, we introduce a new class of collective design system that intermixes people instead of ideas: instead of receiving mere exposure to others' ideas, participants engage deeply with other members of the collective who represent those ideas, increasing engagement and influence. We thus present Hive: a system that organizes a collective into small teams, then intermixes people by rotating team membership over time. At a technical level, Hive must balance two competing forces: (1) networks are better at connecting diverse perspectives when network efficiency is high, but (2) moving people diminishes tie strength within teams. Hive balances these two needs through network rotation: an optimization algorithm that computes who should move where, and when. A controlled study compared network rotation to alternative rotation systems which maximize only tie strength or network efficiency, finding that network rotation produced higher-rated proposals. Hive has been deployed by Mozilla for a real-world open design drive to improve Firefox accessibility.