My intention in posing the question was to limit it to LOTR. Doesn't hurt to talk about The Hobbit, of course.
Eagles, definitely. The King of the Eagles speaks in language to Gandalf.
I wouldn't personally think that Shelob is of the level of conscious intelligence that I was speaking of. Certainly an independent selfhood and an evil will, but it seems to me her observed behavior and Tolkien's description of her actions and intentions seem more animal-like. On the other hand, I've sometimes thought that she might be another of the maiar.
I know there is at least one other example of an animal whose mental processes are related, "inside its head" as it were, such that recognition of another race, memory of prior events, and curiosity about the purpose of actions of other individuals are displayed.
I'm drawing a blank here about a third animal that showed clearly rational thought, that I thought I recalled.
Any further inputs on this one?