Can you legally publish a letter someone wrote to you?
A friend of mine and I have had a little running joke for many years. He says he saves every letter he gets from me so that when I become a rich and famous writer, he can publish them and make a ton of money. This doesn't seem quite fair to me, since, after all, I wrote them, and I deserve as much of the take as he does. He points out that letters from famous people are hot collectors' items, and that nobody asks Richard Nixon, say, if it's all right with him if they sell his letters. Obviously, only Cecil Adams can settle an important legal issue like this one.
A little running joke, eh? Boy, you two do have a lot of fun, don't you?
Collecting and publishing are two very different things. As a material possession--like a pair of socks or a souvenir paperweight--a letter is yours if someone sends it to you, and you can sell it if you like. But as a literary composition, a letter belongs to its author, and only s/he has the right to publish it.
If you acquired the original manuscript of A Farewell to Arms, for instance, you'd have no right to run off a few copies (or, technically, even put it on display--that's considered "publication," too) without the author's permission. The same rules apply to letters as to novels, plays, poems, or whatever.
But I'll think I'll hang on to this one anyway, just in case.