In the T/Z/F/Chi sq test, are we looking at the Distribution of the Control dataset or Treatment dataset? Pretty straight forward question.
Even if we are looking at the shape of the sample's distribution, are we sampling from the Control group or the Treatment group, or both?

We are looking at the distribution of the test statistic.
For instance, in a t-test, we have some parameter we estimate, and we calculate the deviation of the observed parameter from the hypothesized parameter. Then we divide by the standard error. In a two-sample test, that parameter is the difference in means between the control and treatment groups.
When we look at the distributions of the original data, it is to see how well the test statistic will follow the assumed distribution. For instance, in a t-test, the test statistic has the claimed t-distribution when the control and treatment groups are normal. If either group is not normal, then we get a deviation from that claimed t-distribution. Fortunately, the t-test tends to be robust to many violations of this normality assumption (though there might be better approaches, depending on how bad the violation is, and many tests (such as an F-test to compare two variances) have terrible robustness).
