It’s my understanding that the two sons of Arvedui left their father during the capture of Fornost. They were driven westward over the Lune by the King of Angmar’s forces.
Arvedui and some of his guard went north to the North Downs just above Fornost and battled Angmar’s troops. Then Arvedui and his small remnant fled northwest & eventually hid in the old dwarven tunnels at the far northern end of the Blue Mountains.
Aranarth, Arvedui's son, and his group of Dunedain must have reached Cirdan about the same time his father reached the Snowmen. I don't think the king could have used the palantir to signal Cirdan and his sons, since it would require them to have a palantir. It seems to me that Aranarth knew his father was heading north and Cirdan sent a ship with a search party to Forochel to start the search there. It was sheer chance that the king and his men had kept a signal fire on the beach of the Ice Bay and that it was spotted by the sailors.
Quote:
Appendix A; Return of the King:
In 1974 the power of Angmar arose again, and the Witch-king came down upon Arthedain before winter was ended. He captured Fornost, and drove most of the remaining Dunedain over the Lune; among them were the sons of the king. But King Arvedui held out upon the North Downs until the last, and then fled north with some of his guard; and they escaped by the swiftness of their horses.
For a while Arvedui hid in the tunnels of the old dwarven mines near the far end of the Mountains, but he was driven at last by hunger to seek the help of the Lossoth, the Snowmen of Forochel. Some of these he found in camp by the seashore; but they did not help the king willingly, for he had nothing to offer them, save a few jewels which they did not value; and they were afraid of the Witch-king . . . But partly out of pity for the gaunt king and his men, and partly ut of fear of their weapons, they gave them a little food and built for them snow-huts. There Arvedui was forced to wait, hoping for help from the south; for his horses had perished.
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When Cirdan heard from Aranarth, Arvedui’s oldest son, that the king had fled north, he sent a ship to Forochel to seek for him. The winds were contrary and it took a number of days before Arvedui and his remnant were spotted – they had kept a small blaze going near the shore as a signal fire. It was March by then and the ice only beginning to thaw.
The Lossoth were amazed at the sight of the ship. They could not remember ever seeing one in the Bay. They took some of their sliding carts and went out onto the ice as far as they dared with the king and his men to meet the boats from the ship. The Lossoth smelled danger in the wind and counseled the king to let the boats bring them supplies of food and other things so that the king and his little band could wait until summer, when the power of the Witch-king wanes – “. . . but now his breath is deadly and his cold arm is long . . .” the chief of the Lossoth told the king.
Arvedui did not heed the counsel. In parting he gave the chief his ring (the Ring of Barahir), later ransomed by the Dunedain. The Dunedain got on the ship. A great storm of wind with blinding snow from the North came up before they could leave they bay. It drove the ship back onto the ice and piled up ice against it. In the night, the ice crushed the hull of the ship and the vessel foundered. All were lost including the 2 palantiri – the Stone of Annuminas and that of Amon Sul.