Do ions in the medullary interstitium just hang out, or go into the circulatory system and leave the kidney?
You've grasped one of the fundamental principles of the renal physiology -- the medulla must stay salty. Lets start there and see if we can't work out what happens.
As the filtrate flows through the tubule, solutes are removed from the lumen at the thick ascending limb and added to the medullary ECF. This concentrates solutes in the medulla, producing the corticomedullary gradient (the medulla must stay salty). This is called countercurrent multiplication, and explains how such a large osmolar gradient occurs between the cortex and the medulla.
At steady state, with an established corticomedullary gradient, what happens? Solutes continue to be reabsorbed by the tubular cells in the thick ascending limb. In order for it to be steady state, each ion reabsorbed into the ECF needs to go somewhere. One of three things could happen to these ions:
Secretion or diffusion into the collecting duct (and excretion as urine)
Diffusion into the cortex (dissipating the corticomedullary gradient)
Diffusion into the capillaries in the medulla, the vasa recta (exiting the kidney through the renal vein)
In fact, all of these things happen to a certain extent.
Secretion and diffusion into the collecting duct (1) are regulated by changes in the expression of pumps and channels. Diffusion into the cortex (2) is kept to a minimum by diffusion into the vasa recta (3), what we call countercurrent exchange.
I always assumed that those ions were then entering blood vessels for another purpose.
So this does happen, but remember, the main reason for concentrating solutes in the medulla is so that the kidney can, if it needs to, excrete a very concentrated urine. Ions enter the blood vessels for the same purpose, in order to prevent them from diffusing back to the cortex, dissipating the corticomedullary gradient.
The concepts of countercurrent multiplication and countercurrent exchange were proposed in the 1940s and supported by a series of experiments in the 1950s. The landmark 1959 paper by Gottschalk and Mylle in The American Journal of Physiology was reprinted in 1997 by JASN and available for download without a paywall. It's an excellent read! As they predicted, we've learned a lot more, but this model still holds up, and is what we teach in renal physiology to this day.
This is their figure: