The short answer is, historical detective work.

If you know the sender and the receiver, the first thing you normally try to do is see if they're still alive. This isn't the issue in this case, but when August Derleth and Donald Wandrei started compiling what would become the Selected Letters of H. P. Lovecraft, they solicited all of his known friends and correspondents for letters so that they might be copied and returned. If the sender and receiver are dead, however, that's not really an option.

So the next step would be to see if their papers still exist as archives. The more prominent the individual, the more likely this is, although there are no guarantees in life. In the 18th and 19th century, prominent or wealthy families might keep archives of documents - especially legal documents or correspondence - going back decades or centuries. In the 20th century, it is much more likely that such papers if they exist are deposited with a university library, state historical society (in the US), or other historical institutions. After his death, for example, most of Lovecraft's accumulated papers and correspondence were deposited at the John Hay Library of the local Brown University in Providence, Rhode Island; August Derleth's correspondence ended up at the Wisconsin Historical Society, etc.

Relatively recently, a cache of previously unknown letters from Lovecraft turned up in box in someone's house; they were part of his correspondence with a revision client, Zealia Brown Reed Bishop, and had simply been passed down through the family - utterly forgotten until someone recognized what they were - and that's why we now have The Spirit of Revision with the letters between Lovecraft and Bishop. This kind of discovery is exciting, but often coincidental. Diligent researchers sometimes work through the genealogies of the descendants, calling or paying a visit and inquiring after old documents. Doesn't always give results, but sometimes you hit gold dust - several photographs of pulp author Robert E. Howard, a pulp author who had no direct descendants, have come from family photo albums of distant cousins.

If there isn't a publicly-available archive or family archive, things get murkier. A market for historical letters does exist, especially if the individual was prominent in life, or became so after their death; researchers might start haunting bookseller catalogues and auctions looking for a record of letters for sale. Many of Lovecraft's letters to friends were sold after his death, and still pop up occasionally on ebay or bookseller websites; often for very inflated sums. The downside is, such letters are rarely available; the upside is collectors who paid money for them tend to take care of their collections, so such materials are often only "out of circulation" for a lifetime, maybe less if they decide to sell or donate them to a library or historical society.

The nice thing about a library or archive is that they usually maintain some provenance for the materials, so that even if they don't know the ultimate source of the letters, they'll mention when they received them and from whom; this can be very important in trying to trace back up the chain to see who else might still have such letters. If a certain bookseller sold one letter, for example, they might know where more are.

Time and distance, however, make things more difficult. When you're talking about letters a couple centuries old, the possibility space of where the letters could be - if they haven't already been destroyed - is huge, unless you can narrow it down. Finding them either depends on an exhaustive, diligent search and/or a good bit of luck.