What is the supplication that the resident says to the one traveling and vice versa, and should it be recited aloud or silently?

It is recommended for someone bidding farewell to a traveler to say to them: "I entrust to Allah your religion, your trust, and the consequences of your deeds," or "I entrust you to Allah, the One with Whom deposits are not lost," or to supplicate for them with "May Allah provide you with piety, forgive your sins, and facilitate goodness for you wherever you are."

As for the supplication of the traveler for the one remaining behind, it was reported from Abu Hurairah, may Allah be pleased with him, that he said: "I entrust you to Allah, the One with Whom deposits are not lost," and Al-Albani graded it as hasan (good).

It is recommended to raise one's voice with the supplication to an extent that the one being bid farewell can hear.