Background: Existing analyses of nonpharmaceutical interventions (NPIs) against COVID19 transmission have focussed on the joint effectiveness of large-scale NPIs. With increasing data, we can move beyond estimating aggregate effects, to understanding the effects of individual interventions. In addition to effectiveness, policy decisions ought to reflect the burden different NPIs put on the population. Methods: To our knowledge, this is the largest data-driven study of NPI effectiveness to date. We collected chronological data on 9 NPIs in 41 countries between January and April 2020, using extensive fact-checking to ensure high data quality. We infer NPI effectiveness with a novel semi-mechanistic Bayesian hierarchical model, modelling both confirmed cases and deaths to increase the signal from which NPI effects can be inferred. Finally, we study the burden imposed by different NPIs with an online survey of preferences using the MaxDiff method. Results: Six NPIs had a >97.5% posterior probability of being effective: closing schools (mean reduction in R: 58%; 95% credible interval: 50% - 64%), limiting gatherings to 10 people or less (24%; 6% - 39%), closing nonessential businesses (23%; 5% - 38%), closing high-risk businesses (19%; 1% - 34%), testing patients with respiratory symptoms (18%; 8% - 26%), and stay-at-home orders (17%; 5% - 28%). These results show low sensitivity to 12 forms of varying the model and the data. The model makes sensible forecasts for countries and periods not seen during training. We combine the effectiveness and preference results to estimate effectiveness-to-burden ratios. Conclusions: Our results suggest a surprisingly large role for schools in COVID-19 transmission, a contribution to the ongoing debate about the relevance of asymptomatic carriers in disease spreading. We identify additional interventions with good effectiveness-burden tradeoffs, namely symptomatic testing, closing high-risk businesses, and limiting gathering size. Closing most nonessential businesses and issuing stay-at-home orders impose a high burden while having a limited additional effect.