The need for public engagement is increasingly evident as discussions intensify around emerging methods for carbon dioxide removal and controversial proposals around solar geoengineering. Based on 44 focus groups in 22 countries across the Global North and Global South (N = 323 participants), this article traces public preferences for a variety of bottom-up and top-down engagement practices ranging from information recipient to broad decision authority. Here, we show that engagement practices need to be responsive to local political cultures and socio-technical environments, while attending to the global dimensions and interconnectedness of the issues at stake. Establishing public engagement as a cornerstone of inclusive and sustainable governance of climate-intervention technologies requires (i) recognizing the diversity of forms and intensities of engaging, (ii) considering national contexts and modes of engagement, (iii) tailoring to technological idiosyncrasies, (iv) adopting power-sensitive practices, (v) accounting for publics' prior experience, (vi) establishing trust and procedural legitimacy and (vii) engaging with tensions and value disagreements.