Isaiah had met, it seems, privately with Ahaz, at the aqueduct, warning him to trust in the Lord and offering him a sign. When Ahaz refuses, Isaiah gives him a sign nonetheless. There is no repentance on Ahaz's part, and so Isaiah 8:1-2 seems to follow the model of discipline set forth by Jesus later in Matthew 18. If a man won't hear you privately, take two or three others, and then finally, lay it out publicly.
The public declaration of the name, "Speed the Spoil, Hasten the Booty," and the fact that this child will be born and before he grows up will see his name fulfilled upon Syria and Israel make this prophecy-in-action a very public one. And of course, this is not because God is a very good predictor of the future, but the One who speaks and it is so.
But there is something else going on from "private" to "public." Ahaz had been instructed and warned as the king of Judah and as an individual who had to determine whether or not he would trust God. He refused. Now the same warning is given to the people (8:11). They end up following their king (not King Yahweh, but King Ahaz) with their own disastrous response.