Question: Does it make sense to compare the sensitivity and specificity of a screening test between two groups? The two groups underwent the same screening test. Does it make any sense to compare the calculated sensitivities and specificities betweeen the two groups statistically?

Answer: 
It might, but if the data came from a prospective cohort study you would be acting as if the sampling were retrospective, since sensitivity = Prob(past | future) = Prob(test + | final diagnosis +). Also sensitivity and specificity are not constant but vary with subject characteristics. This is delved into in detail in [Chapter 18](http://biostat.mc.vanderbilt.edu/tmp/bbr.pdf "Chapter 18").
It definitely makes sense. It might tell you whether there are meaningful differences in the performance test in light of the two group features.
I would also compute positive likelihood ratio, negative likelihood ratio, diagnostic odds ratio, and area under the curve of the summary receiver operating curve (SROC).

I have recently conducted a meta-analysis on diagnostic accuracy of myocardial perfusion scintigraphy, and you can compare the diagnostic performance of this medical test in two groups of patients with different types of features by means, for instance, of meta-regression or statistical interaction.

This can also be explored graphically, e.g. building two superimposed SROCs with the [mada](https://cran.r-project.org/web/packages/mada/vignettes/mada.pdf) R package.

The same can apply to single studies.