How can digital signatures assure the sender their message has been correctly received?

PKI ensure that if the message reaches its destination it has not been altered (if signed with sender private key) and/or has not be compromissed (if crypted with recipient public key).

If the sender wants to make sure that the recipient has actually received the message, a higher level protocol must be used. For example the recipient could send a signature of the original (optionaly decrypted) message using their private key. So if the round trip:

A -> message signed with A private key -> B
A <- signature of the original message with B private key <- B

completes, then A can be sure that B has received the correct message.

If you do not set up that round trip, even with modern system where the sender can be sure that the message reaches the recipient system, you have no protection against the message being destroyed between its arrival on a machine and the moment when the human being named B could read it.

This would be more or less an implementation of what was the QSL in the early days of radio frequency message (mainly using Morse code). BTW that QSL was still in use in the 80’s to ensure that a message had been received and understood (*): until the sender had not received a QSL to message number xxx they periodically try to send it again or try a phone call to know whether the recipient system was off or out of use (at least at French Met Office).


(*) as QSL to message … had to be manually sent, it meant that somebody could read the message number and declared having understanded the full message.