The Lancet has published an impressively detailed study of COVID transmission in UK schools when they re-opened certain grades in reduced-risk settings from June 1 to July 17. This includes detailed analysis of each case, with a level of reporting and investigation that is an order of magnitude beyond what the US or any state is capable of doing. Very, very well executed.
tl;dr - If you reduce class sizes dramatically, keep student/teacher groups strictly bubbled, and test/trace effectively, resuming in-person schooling is a fairly low risk process.
They only brought back certain grade levels, and only in areas with low transmission in the community, so overall their COVID rates were about 100x lower than the US is seeing now. (Yes, per population. This happened during their summer low, which was much lower than our best weeks.) This allowed them to identify a lot of asymptomatic cases in students through contact tracing in families, and to remove those students before they could pass on the infections.
They also "bubbled" the students into smaller-than-normal groups that stayed together with better distancing than is possible in normal density classes. This is very different than the current US situation, where students in grades 5+ are often exposed to over 100 students per day, so the results are not directly comparable to full-scale opening of in-person schooling.
With those caveats aside, this is convincing data that classroom instruction can be done safely with small groups that stay bubbled together. The outbreaks that did happen showed that the risk of transmission from and between staffers was 2-3x higher than the risk from students. Staff-staff transmission happened a lot, child-child transmission was relatively rare, even in the year 10 and 12 (high school) cohorts.
Because of their extensive testing, surveillance and reporting systems, they were able to rapidly isolate contacts when cases occurred, so most resulted in no transmission in the school, and the ones that did were mostly limited to one additional case.
Staff ended up having about 3x the level of cases of the general population, which is probably not unusual among "essential workers" that can't work from home.
It would be really interesting to see a similar study with students having 5-10x more contacts as they do in US schools that don't keep classes isolated through the week.