For the Pfizer and Moderna vaccines, efficacy is commonly reported as something high (like 95%) after two doses, and something lower (in the 50-80% range) after only one dose. However, I think there's good reason to doubt that this 50-80% number is capturing the right thing, and there's good reason to believe that the mRNA vaccines are ~95% effective starting 12 days after the first dose. This is important because if your immunity is as high 12 days post-first-dose as it will be 12 days post-second-dose, then any precautions that you are waiting until you are fully vaccinated to get rid of can instead be shed 3-4 weeks earlier.

(Note that getting a second dose might still cause your immunity to wane more slowly in the long term; that is, second doses might still increase long-term efficacy.)

Overall, my current best guess, with confidence ~65%, is that the mRNA vaccines are equally effective 12 days after the first dose as they are 12 days after the second dose.

Below I'll explain the cases for and against the claim that the second dose boosts immunity. In short, the phase 3 data strongly suggests that it doesn't, and real-world data from Israel strongly suggests that it does. I'm posting this as a question because I hope that someone who knows more about this, or who has seen data that I haven't seen, can help figure this out.

Why might we think that second doses don't boost efficacy?

The short answer: because this is what the data from the phase 3 trials straightforwardly says. Here are the key graphs; note the lack of apparent effect from the second dose.

So what's up with the commonly reported 50-80% post-first-dose efficacy? The issue is that this number captures efficacy starting on the day you receive the vaccine (or sometimes 7 days later). As you can see from the graphs, is too soon for the vaccine to have any effect at all. Rather, immunity starts appearing in the data starting sharply on day 12, so what we really want to know is the efficacy in the time period between day 12 and the second dose.

(Note that since there's an incubation period between exposure and developing symptoms, the efficacy actually starts before day 12, perhaps around day 7. Shouldn't there be some noise around the day the vaccine kicks in due to varying incubation periods? I would think so, but we don't see that here.)

Frustratingly, the phase 3's don't report this number. But using some data included in the Pfizer phase 3, I was able to make this graph:

(They only give data binned by week, so I can't make this graph on a day-by-day basis. Also, it gets noisier as the trial continues because the number of participants dropped, so that oscillation after 70+ days is probably nothing real. Alas, I can't find the data in the Moderna phase 3 necessary to make this graph for Moderna.)

 So based on the phase 3 data, it really doesn't look like the second dose does anything to boost immunity.

Why might we think that second doses do boost efficacy?

Because that seems to be what Israeli data is saying. Here is the key graph:

Unlike the graphs above, it actually does take some time for the blue line to flatten out. This is not just an illusion; see here:

It's important to note that the Israeli trial is about 30x larger than the Pfizer phase 3 (N = 43K vs N = 1.2 million). 

But it's equally important to note that the the phase 3's were true randomized controlled trials - the placebo group was given a fake injection and everything - whereas the Israeli studies were observational. So it's possible that something funky is going on, like that once you're vaccinated (and you know you're vaccinated) you go out partying in Tel Aviv and get Covid at higher rates than whichever non-vaccinated control person the study matched you with; then over time the excitement wears off and you start going out partying at close to population base rates.

One thing I'm uncertain about is how to weight the relative evidence from the phase 3's and the Israeli data, given that there is much more Israeli data but the phase 3's are RCTs (and are still quite large). For now, I'm trusting the RCTs slightly more, so my 65% confident guess is that the second dose does not boost your immunity.

But I would love to hear from others.

Sources: Pfizer pfase 3, Moderna phase 3, Israeli data