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The vaccines being distributed in the US and EU have both completed _three_ stages of trials run under quite strict conditions (the only change being that parts of these trials were run in parallel instead of sequentially - see fig 1 from [1]).

People getting sick after only receiving the first shot is fully expected. The efficacy of the Biontech vaccine was only 54% after the first shot and only rose to 94% after the second. And notice that it is not 100%.. so you would still expect some people to get infected after the second shot.. just much fewer cases.

[1] https://www.nature.com/articles/s41586-020-2798-3



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