Can I compare p-values? Assume I have 2 vaccines A and B. They are tested on 2 groups of patients. Say each group has n=1000 people drawn from the same population, vaccine A is used on group 1 and vaccine B is used on group 2. It is observed that in group 1, 70 male patients and 40 female patients are cured, while in group 2, 55 male patients and 50 female patients are cured.
I can then test the following hypothesis:

*

*vaccine A is more effective on male patients than female

*vaccine B is more effective on male patients than female

Each hypothesis test will produce a p-value, p1 and p2. Now I see that p1 is smaller than p2, is there anything I can say regarding which vaccine is more effective on male patients?

Since effect size is one factor going into p-values, there are cases where p-values tell you something useful about effect sizes, but just looking at effect sizes directly is always more useful.
You have the further issue that your p-values are from testing the hypothesis of the vaccine being more effective on males than females, rather than directly testing effectiveness on males. This means that your p-values are differing not only from the variable you care about (effectiveness on males) but also on one you don't (effectiveness on females). This makes the p-values even closer to useless.
There would be little reason to not simply do hypothesis testing based on a binomial or Gaussian hypothesis of the data. If for some reason you were only given the p-values and did not have access to the raw data, the best course of action would probably be to simply refrain from trying to draw conclusions.
As as a purely medical, rather than statistical note, generally vaccines are used to prevent infection rather than cure them (although I suppose there are cases where they can do the latter).
