Once someone with diabetes insipidus begins treatment are they able to resume a diet with moderate sodium (after being on a very low sodium diet)?

I'd still restrict. : Di results in inappropriate water loss in the urine. As long as you have free access to water, you shouldn't run into any serious problems, but even with specific treatment, modestly restricting your salt and protein intake can lessen the thirst sensation and subsequently the volume of urine you will make. That means less trips to the bathroom.

DI and sodium: There is no reason that one could not resume a moderate sodium diet after di is treated. Serum sodium is a measurement of total body free water much more than a measurement of total body sodium.