Despite allusions to a washroom/bathroom at the Prancing Pony in the Fellowship and perhaps some other places I cannot think of at the moment, I would think that pondering too much about these things would take away the bliss of Tolkien's story. I mean, we KNOW it is there...especially among Men and Hobbits. Elves? Who can say? They ate otherworldly food (lembas) given to them by the gods, healed quickly, were generally mightier and hardier and wiser than the other races, so who is to say Elves went to the bathroom like mortal Men?
Not that exposing Middle Earth's earthiness would necessarily take away, (hobbits are particularly anyhow) but dwelling on these things would certainly take away from the high tone that Tolkien sets ("...and then Frodo slipped away to use the toilet. Now a hobbit toilet is quite like ours is today...") Well, MAYBE...if it was describing something ABOUT the toilet that was relevant to Hobbitdom...not just for the sake of saying , "By the way, Hobbits poop too and here is where it goes to in the Shire..." Not for the sake of satisfying (I apologize ahead of time) some of our more baser and rational curiosities.
Tolkien does delve into such things as the intimate relations of Elves in other writings, and this is almost too much (in my small opinion); it comes close to bringing Elves, particularly Elves, down to a human level when they are meant to be representative of humanity before a Fall; as such, higher beings. Allusions leading us to dwell otherwise, even briefly, would certainly be in the realm of Potty: of describing in too much detail, bathroom, sewer, and waste. By implication, certainly it was THERE in many places...most certainly I could imagine it with Hobbits, Dwarves... But even alluding to one of the Eldar dropping a load and then describing where it goes? There is some rule that staunchly disallows this on the tip of my tongue, yet I cannot think of what it is called at the moment...we all have wondered such things as readers though.