I started doing this long ago to refresh my German in order to pass a language proficiency exam when I started my doctorate. Since I know the text of LOTR so well, if I get bogged down in the German, I can recall the English almost word-for-word and figure out what's going on. My German vocabulary is improved thereby, but also my knowledge of sentence construction and idioms auf Deutsch.
And I notice another benefit. In reading the descriptions and dialogue surrounding Gandalf and Denethor, both characters emerge more powerfully. Denethor is much more the great lord, and Gandalf's personality is equal to the Steward's. The fine shades of language in their verbal fencing comes through, perhaps because I'm paying more attention to the actual words used.
Tolkien was a master of using English to portray variations in age, status, culture, education, and even species. But to an English reader, especially one who started reading LOTR so young, some of this goes right past one; some of the higher status characters begin to sound alike, and we miss the cues that distinguish (for instance) the very ancient Elrond from the merely old Theoden and the prematurely aged Denethor. Somehow, I notice all this more in German. Not because German is a more flexible language, though certainly it retains a lot more of the antique forms than English does. I think it's just because I'm paying closer attention.
No doubt if I were talking to Germans about reading LOTR, I'd be saying that you really can't "know" the work unless you meet it in its original language -- an argument that Tolkien made concerning Beowulf, for instance, and that others have made about Homer and Virgil and Dante. But that doesn't mean that a translation isn't sometimes very good and brings you pretty close to the living spirit of the original author.