What methods have you been taught to make such distinction?
But I don't want to look at means when the sets are nonparametric do I? They are not normally distributed
The test is robust, even when the variables are not normally distributed. Second, you have 1000 samples, by the central limit theory, the difference in the means can be modelled by the t-distribution. Third, you can check whether your data is skewed or not by looking at their histogram or dot-plot. If it does not show extreme skewness, it's safe to make that inference.
As you have more samples, it should look more normally distributed. We can't determine if it's too skewed for 1000 samples based on 200 samples.
If I'm trying to be statistically careful though, is it really good enough to just visually look at and say it's normally distributed enough? I need to write up an official document with these results so I feel like there must be a statistical normality test I can use instead of just eyesight. When I ran a D’Agostino/Pearson’s normality test on these datasets, it was very non-Gaussian (P<0.001).
That is up to you. Your responsibility is to defend your methodology and findings as there isn't one correct way to do statistical research.