While the receiver of a letter on paper certainly owns the letter she doesn't own its contents. The copyright of the letter remains with the author.
So you can frame the letter and hang it on your wall you violate the authors copyright if you publish it without authorization.
What you can do is sell the physical object. That belongs to you.
You can also burn it or destroy it - which is an interesting edge case.
This really comes down to different kinds of ownership. The whole point of copyright is that creative/cultural/intellectual property rights are a thing, analogous to tangible property but with a different set of legal and commercial rights and obligations.
Electronic media depend on context. ISP and telcos are usually considered common carriers so they don't own the copyright of emails or SMSs.
Social media networks either claim all copyright in full (bad news for you...) or have a specific license clause in the T&Cs which gives them reproduction rights.
Sometimes these are for specific limited purposes, and sometimes they're a blanket sign-away of all exploitation rights. (Bad news for you again.)
Also, how does it work if a whistleblower gives a journalist letters that they received from someone else? Is the journalist not allowed to publish them in an article?
To the down-voter: could you please elaborate with what you got the problem so we could potentially discuss it?
If I send you an email, same thing, if it survives for such a long time. Does Google/any ISP also own it?
If I send you a SMS, same thing, I bet it won't survive. Does the phone company also own it?
If I write you on a social network, do you and I own the message or does the social network own it? There are ToS that we might not have read.