https://www.medrxiv.org/content/10.1101/2022.03.20.22272571v1.full.pdf
For first infections, 19% of cases were associated with hospital admissions.
For second infections, 17% of cases were associated with hospital admissions - not a very significant difference.
For third infections, 25% of cases were associated with hospital admissions.
The authors also found an 8x increase in reinfection during the beginning of Omicron, from November 2021 to January 2022. This was the highest raw increase in reinfections, but the highest rate was just at the beginning of the Delta wave in Spring, 2021.
The authors note that this data is a lower bound.
So not only does reinfection not give you protective immunity, it actually increases your chance of catching it again, and getting a more severe infection. This means that not only is the herd immunity strategy pointless, but it's also actually actively making things worse over time compared to not doing it. Shitty both in the short term and long term.
https://www.medrxiv.org/content/10.1101/2021.07.22.21260972v2.full.pdf
This study, combining 81 different studies in 22 different countries, reported a 3x increased hospital admission rate on reinfections, and a doubling of the number of patients who needed oxygen.
I don't know for sure. To give a guess, I'd imagine that vaccinated people would do better. But I'd also say that vaccination would only really matter if it was ~6 months ago, so as long as you're keeping yourself topped up with boosters, then you should be fine. As well as mask wearing and all that to decrease your odds too.
Nature recently found that, in the context of BA1 vs BA2, "a study published on 13 March shows that mRNA vaccines offer a similar degree of protection against the two strains — although protection against SARS-CoV-2 infection and symptomatic disease wanes within months of a third dose." The timescale is that after 4-6 months, there's an average of 10% protection remaining. However, that's only overall - if you consider just severe symptoms then the vaccines give you better protection for much longer: "This analysis showed that protection against severe disease remained at 68% or higher for at least 7 months, even in people who had only received two vaccine doses, and shot up to over 80% after a booster dose."