If you had an address, they sent a messenger out with a copy of the telegram to deliver it (in America, tip expected). This is the person who later delivered the candygram or the singing telegram. If the telegram was sent w/o an address, other than that of the telegraph office, it was a "will call" - that is, they expected you to drop in (call) for your messages, same as the PO. Of course, your hotel would take your telegrams, same as your mail.

In the age of the telephone, if given a number, they would phone you and read it to you. This might be cheaper than a long-distance call, and the only way for someone contacting you from a place w/no phones, but a telegraph. (Say, Senegal to Oshkosh in 1932).