The sum of the means of two normally distributed random variables is the same as taking the mean of the sum of the two signals.
Does this hold true for maximum likelihood estimation?
Is summing the estimated means of two given data sets by maximim likelihood method, going to give me the same result as finding the mean of the sum of the two data sets with the maximum likelihood method? And the same question for the variances
What is true for the mean which is a linear function of your data is not true for the variance which is not a linear function of your data.
So, summing your data to find the mean of the sum with a Maximum Likelihood (ML) method can work but then you have to give up on the variance.
If you want to perform a unique ML algorithm for your two data sets, maybe you could have a look at the ML for mixtures of Gaussian distributions and adjust the weights with respect to your data sets sizes. This way you will be able to get all the information and then add up the mean if you want.
Empirically however and if working with signals that are "not truly Gaussian", if you are only interested in the final result (the sum of the two signals) and do not really care about the variance, it is possible that in some cases you get better results by summing the signals beforehand. This is because any ML estimation will have some estimation error but also a risk to converge towards a local extremum (which is not the global one). So in some cases, it can give better results for the mean, but only experiments can show this.
