To me the claim is ambigous in that it may mean
incidence among vaccinated is 1/10 of the incidence among unvaccinated,
this is true in the sense that one can find a places and times where it is/was true. But it is nothing fixed, this ratio changes.
Probabilities to get infected [by the same index patient] are 1 : 10 comparing vaccinated : unvaccinated high risk contact persons.
(My guess is that this is meant, since she goes on with probabiliy of onward transmission)
Ratio in incidences vaccinated : unvaccinated
Possibility 1. may be true at a particular point in time for a particular population (e.g. Australia right now - but I didn't find any data that would confirm this) - but this would still be subject to substantial change e.g. if behaviour changes between vaccinated vs. unvaccinated subpopulations (e.g. more or less NPI are mandated or customary for vaccinated)
E.g. the German Land (state) of Thuringia reported a factor 10 between notification rates for unvaccinated / vaccinated beginning of September. After that, the ratio decreased to about 3 by mid November:
(they give 2 caveats that both bias the notification rate among vaccinated downwards: a) vaccinated are counted as case only if they have a positive test and symptoms, but unvaccinated are counted with positive test regardless of symptom status. b) They say that vaccinated are likely to be tested less frequently than unvaccinated, which can lead to underestimating case rates among the vaccinated. [However, this would largely be among asymptomatically infected vaccinated - who'd be discarded from counting
anyways according to a). So bias b) would be part of the bias a) ]
OTOH, since only symptomatic cases of infection are counted among the vaccinated, we can also say that the likelihood of getting covid (as in not only being infected, but also having symptomatic disease) cannot be higher than the ratio between the vaccinated and unvaccinated lines here. So also no factor of 10 in that respect.
Probability of transmission/infection
Point 2. would be overestimating vaccine efficacy against infection:
Lambi argues "We do it to protect those who can’t protect themselves", which I read as those at high risk of severe or fatal course if they get infected. Thus, we need to look at vaccine effectiveness against transmission, not only at preventing for the vacinee a severe course.
Vaccine effectiveness against infection and onwards transmission of COVID-19: Analysis of Belgian contact tracing data, January-June 2021, reports the following reduction in transmission compared to unvaccinated index patient - unvaccinated high risk contact person:
That is, depending on the vaccine used the probability of getting the virus (as in: the high risk contact person has a positive PCR test) is very roughly somewhere between 1/2 (Astra, Johnson) and 1/7 (Moderna) compared to an unvaccinated high risk contact.
The effectiveness against infecting someone else is lower, though:
A vaccinated index patient is anywhere between about as likely (Astra) and 60 % less likely (Pfizer) as an unvaccinated index patient to infect an unvaccinated high risk contact.
These findings may be somewhat biased towards too low vaccine effectiveness if asymptomatic cases are unlikely to be discovered (i.e. do not become index for contact tracing) and also infect fewer people (which is plausible, but so far I've only seen surrogate data about this)
The best point estimate is Moderna with 1/7 probability of getting infected ⋅ 1/2 the probability to pass it on, a total of 1/14 rather than the 1/30 claimed. Pfizer is at 1/10, Johnson about 1/3 and Astra somewhere between 1/2 and 1/3 in total.
Findings for Delta
During the Belgian study the original, then alpha and delta strains were predominant.
The CDC has a report about vaccine efficacy against infection that distinguishes alpha and delta, giving VE for Moderna and Pfizer against delta of 53 %. They do not distinguish probability of being infected vs. infecting, the 53 % is the total reduction (diagonal of the matrix above).
A UK preprint reports:
This is a reduction in onward transmission of delta somewhere between 40 % (fresh Pfizer) and 0 (Astra 3 months later) - a rather meagre effect compared to the protection one gets by being vaccinated oneself.
Note: these reductions are in the same order of magnitude like other measures, so can rather easily be (over)compensated by dropping other measures.
E.g. meeting outdoors vs. indoors reduces the transmission probability to about 1/18. I.e. for the question of ongoing transmission, someone meeting only outdoors is as good as a Moderna vaccinated in the Belgian study who only meets indoors. Doing every 7th or 8th meeting outdoors is about as good as someone vaccinated with Astra (Belgian estimate) or mRNA during delta predominance (CDC estimate) who always meets indoors.
(Vaccination of course has the advantage of working for many meetings, and it gives additional protection against a severe course to the vaccinated.)
The difference in infecting someone else if you are infected and vaccinated vs. not is in the order of magnitude of having your mask fit better or not so well.
To judge the risk for (and protect) people who "cannot protect themselves" we need incidences in addition to the probability of onward transmission
All in all, we have to expect that while vaccinations help, they are not sufficient to guarantee low incidences in the vaccinated population. (For an example, see Ireland right now - who have a notification rate of > 600 new cases per (100k pop ⋅ week), despite 89 % of the population aged 12+ [or 75 % of the total population] being fully vaccinated. The case rate quadrupled within maybe 6 weeks.)
We can formulate this also as: delta has an R0 of about 5. In a somewhat handwavy back-of-the-envelope calculation, we can say reducing the transmission probabiliy to 1/R0 would get us into a barely stable situation. We'd get this after vaccinating p = (1 - 1/R0) ⋅ 1/VE. With the CDC estimate of 53 % VE against infection (transmission) of delta for the mRNA vaccines, we'd need to vaccinate about 150 % of the total population - so that's plain impossible.
Even with vaccination, we're thus in a situation where R > 1 and thus sharply increasing case rates can easily occur.
The risk for "those who can't protect themselves" would thus be much better judged by monitoring the case rate (incidence) among vaccinated and unvaccinated (separately; the incidences also account for any effects due to otherwise different behaviour or environmental factors that may be different between vaccinated and unvaccinated) and then looking at the probability of onward transmission of vaccinated and unvaccinated index patients. Which according to the UK findings for Delta at best are avoiding 2 out of 5 onward transmissions, maybe only 1 out of 10 (3 month "old" Pfizer).