Legitimate question: How do these governments function? Not just facing the challenges of being divided like this, but in general?
You are completely correct. The fact of the matter is that its impossible for a state to centralize with such chaotic borders and non continuous states, to the angst of Alexander Hamilton. As a result, both states in both countries are pretty autonomous and really cannot be forced to do anything. With the Confederation at least the Articles already provide a template for governing. For the Republic that is much trickier, although they have the benefit of access to ports and being connected in the Mid Atlantic and New England. Its easier for the Republic to govern Maryland than Georgia for example. However the Republic still have to deal with all the growing pains that the United States did in OTL although at least this is somewhat easier with a more homogeneous population.
As it is right now they largely work together because they share a common interest in having a functioning central government especially with a resurgent
Western Confederacy. But a state can easily go their own way and neither the Republic or the Confederation can really force a state to do anything. The Republic lacks the power and the Confederation lacks the power and the constitutional authority.
It's also worth mentioning that the borders between the Republic and the Confederation are porous and people can easily travel and trade. That is really important for trade and commerce so the economy of both countries aren't hindered. But really all the borders signify is where the authority of one consitution ends and another begins.