I found these words from Frank Sheed very illuminating:
"A son is a distinct person from his father; there is no way in which a father can be his own son. But though they are distinct persons, they are like in nature — the son of a man is a man, of a lion a lion. In this solitary case, the Father’s nature is infinite; so the Son too must have an infinite nature. But there cannot be two infinite natures — one would be limited by not being the other and by not having power over the other. Therefore, since the Son has infinite nature, it must be the same identical nature as the Father’s.
This truth, that Father and Son possess the one same nature, might remain wholly dark to us if St. John had not given us another term for their relation — the second person is the Word of the first. In the first eighteen verses of his Gospel we learn that God has uttered a Word, a Word who is with God (abiding therefore, not passing in the utterance), a Word who is God; by this Word all things were made.
So God utters a word — not framed by the mouth, of course, for God has no mouth. He is pure spirit. So it is a word in the mind of God, not sounding outwardly as our words sound, akin rather to a thought or an idea. What idea produced in God’s mind could possibly be God?
Christian thinking saw early that it could be only the idea God has of himself. The link between having a son and having an idea of oneself is that both are ways of producing likeness. Your son is like in nature to yourself; your idea of yourself bears some resemblance to you too — though it may be imperfect, for we seldom see ourselves very clearly; too many elements in us we see as we wish they were, too many we do not see at all.
Are we venturing too far if we feel that God does not have the idea for the sake of information about himself, but for the sake of companionship. However this may be, the idea that God has of himself cannot be imperfect. Whatever is in the Father must be in his idea of himself, and must be exactly the same as it is in himself. Otherwise God would have an inadequate idea of himself, which would be nonsense. Thus, because God is infinite, eternal, all-powerful, his idea of himself is infinite, eternal, all-powerful. Because God is God, his idea is God. “In the beginning was the Word, and the Word was with God. And the Word was God.”
So far, the reader may feel that all this is still rather remote — full of significance, no doubt, to theologians, but not saying much to the rest of us. With the next step we take, that feeling must vanish. The Father knows and loves; so his idea knows and loves. In other words the idea is a person. Men have ideas, and any given idea is something. God’s idea of himself is not something only; it is Someone, for it can know and love.
The thinker and the idea are distinct, the one is not the other, Father and Son are two persons. But they are not separate. An idea can exist only in the mind of the thinker; it cannot, as it were, go off and start a separate life of its own. The idea is in the same identical nature; we could equally well say that the nature is in the idea, for there is nothing that the Father has which his Word, his Son, has not. “Whatsoever the Father has, that the Son has in like manner” (John 16:15). Each possesses the divine nature, but each is wholly himself, conscious of himself as himself, of the other as other."