Understanding the impact of different social interactions is key to improving epidemic models. Here, we use extensive registry data -- including PCR test results and population-level networks -- to investigate the impact of school, family, and other social contacts on SARS-CoV-2 transmission in the Netherlands (June 2020--October 2021). We isolate and compare different contexts of potential SARS-CoV-2 transmission by matching pairs of students based on their attendance at the same or different primary school (in 2020) and secondary school (in 2021) and their geographic proximity. We then calculated the probability of temporally associated infections -- i.e. the probability of both students testing positive within a 14-day period. Our results highlight the relative importance of household and family transmission in the spread of SARS-CoV-2 compared to school settings. The probability of temporally associated infections for siblings and parent-child pairs living in the same household was 22.6--23.2\%, and 4.7--7.9\% for family members living in different household. In contrast, the probability of temporally associated infections was 0.52\% for pairs of students living nearby but not attending the same primary or secondary school, 0.66\% for pairs attending different secondary schools but having attended the same primary school, and 1.65\% for pairs attending the same secondary school. Finally, we used multilevel regression analyses to examine how individual, school, and geographic factors contribute to transmission risk. We found that the largest differences in transmission probabilities were due to unobserved individual (60\%) and school-level (35\%) factors. Only a small proportion (3\%) could be attributed to geographic proximity of students or to school size, denomination, or the median income of the school area.