Error for combining multiple binomial distributions

#1
Lets say we have two processes and we want to know which has a higher success rate. So we do two sets of trials. process 1 gives 53 out of 606 and process 2 gives 32 out of 595. So

p1 is 0.0538 with a 95% CL of [0.0371,0.0751]
p2 is 0.0875 with a 95% CL of [0.0662,0.1128]

It would seem that p2 has a higher success rate. However, both only represent one measurement of the success rate and in general these experiment are on different samples. To get the actual success rate we would need to perform multiple experiments of this type and then calculate the mean of the p1s and p2s. My question is how do I calculate the error on this mean? Is it calculated from the distribution of p1 and p2 or do I need to incorporate the confidence limit for the binomial distribution of each experiment in some way? I am happy to stay in a regime where we can make a nearly Gaussian approximation like with the numbers given above. Thanks in advance.
 
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#2
I am going to add the partial answer and assume it is not possible to fully answer this question. There is a standard way to combine several experiments if they can be assumed to have a standard error on the measurement. If the set of measurements is ai and the set of associated errors is σi, then the estimate for the true a is given with accuracy σ by the following:

a = Σ(ai/σi^2)/Σ(1/σi^2)

1/σ = Σ(1/σi^2)

This just amounts to a weighted sum. The reason this does not work in general for a binomial experiment is that there is no way to calculate a σ for each measurement that makes sense. In low statistical cases like 1 success out of 4 the error is highly unsymmetrical.

However, if we are in a situation with high statistics like I gave above we can use something like the Wald interval. One can then only combine them confidently when each individual measurement is similar and derived from high statistics.