For the most part, those who died on campaign, would, die and be buried near the place they died. Roll-call came the morning after the battle and those who did not answer roll call were assumed dead or missing. Missing could mean they were either captured or deserted.

In ancient times, the Romans actually had a mail system and there is evidence of Roman soldiers from across the empire sending mail to their families. This mail would be written on pieces of wood (thin pieces from the ring of the tree) and would be packaged together along with mail from other soldiers. Check out the Vindolanda tablets as examples. In doing so, the Roman Empire had a way to tell families that their sons had died (most soldiers who died until quite recently died from disease rather than from battle).

However, after the collapse of the Roman Empire, the idea of being able to tell family that their sons had died vanished as well. “Conscript obligation” (рекрутская повинность) during the Russian Empire went from life-time conscription to 25 year conscription. The length of service was long, and rarely did a serf return home to simply visit their family. The families generally assumed their son would die (usually from disease) or if their term finished, would settle near where the garrison was located. So, the families would never find out if their son had died... ever.

It's hard to imagine that the idea of families finding out about the deaths of their loved ones would be rendered unimportant after the fall of the Roman Empire. This was a major part of society for many Empires up until the modern-mail system was created across Europe starting in the 18th century. Until that point, someone had to actually go home and relay a message in-person... and if the son lived thousands of miles from the place where the battle occurred (just look at the size of the Russian Empire and you can start seeing how difficult that would be) then, the chance of someone actually visiting the house and telling a family of what had happened to their son was basically zero.

Newspapers were one way of telling families that their loved ones had died. Starting in the 19th century, newspapers would have weekly columns listing the soldiers killed in combat. Hard to imagine families turning the pages to check if their son or father had died... but that was the system back then. Note, the bodies would very rarely ever be sent back home. Burials would almost always take place near the site where the person died. This has only recently changed (starting around the Second World War and the innovation of modern-day graves registration units).