If you are very famous, it’s safe to assume that your letters will someday be published. Under the circumstances, they may have been written with at least one eye flirting with posterity, meaning a conscious aim at being publishable as literature. But if you are also modest, you will write simply and directly for the addressee alone, with no thought of literary effect. Yet even then, given who you are, you may still produce missives of historical, philosophical, psychological, and, however unintended, autobiographical interest.
Let me say of Samuel Beckett that, to his credit, his correspondence exhibits no such intellectual coquetry. The Letters of Samuel Beckett—in four hefty volumes, of which the fourth still awaits publication and the third is just out—are totally spontaneous and unpretentiously intended only for their recipients.