Moderna's third stage trial consisted of 30,000 recipients (presumably that means 15,000 since half of them would have received a placebo.) Added with the previous two trials, you'd expect even the rarest reactions to start showing up already.

You can expect that if there are indeed super-rare novel responses, they'll come to light over the approximately year-long roll out.

Additionally, even though vaccines will be rolling out, Moderna is holding 88 trials over the next 2 years. So they're not just ending the trials and injecting everyone they can.

That's just the one company. Pfizer and the others should be similar.