"No net effect" = tears are produced by the lacrimal glad. I am more of an interested hack, compared to someone that actually knows something (medical professional like a neurologist or ophthalmologist), but my educated guess is that cells in the lacrimal gland produce tears entirely from inputs from the circulatory system, and because of the circulatory system structure in your brain v. face, it's actually rather not as local as the physiological proximity suggests. So the change would be indirect, mediated by blood, and since the concentration of sodium is the same in each crying wouldn't effect blood sodium concentration in the first place. You may under-appreciate how rapid blood moves through the body, too, and quickly any local change would move to equilibrium.

ps: you're welcome, happy studying! (it's daunting, keep in mind https://news.ycombinator.com/item?id=1390027, and remember that some of the smartest people out there become doctors, and they spend a dozen+ years training in just single areas of study and indefinite continuing education)