I (the OP) add here, as an answer, the conclusions I've arrived at after studying the subject deeply.
Personal beings will according to their nature. This implies that all human beings have a unique personal nature and a shared sin nature. This idea seems to help us understand how Christ could have both divine and human natures.
It also suggests a solution to the dilemma of us living eternally in the next kingdom, yet not sinning; as well as our becoming "partakers of the divine nature" (2 Pet 1:4) -- the sin nature that was bound to Adam's progeny is removed and some part of the divine nature replaces it, but the personal human nature is not; in this manner we become deeply "Christ-like" with two natures, human and divine.
Further, it sheds light on Paul's "war of the wills" in Romans 6, which is a war between what we, as redeemed individuals, will (to be like Christ) and what our "old man", i.e. our sin nature, wills (to be independent of God).
Nature and Person
For many, the doctrine of the Trinity is confusing and seems contradictory because in commonplace expression it has been misstated. It therefore seems to be a contradiction in mathematics, the objection being that "three cannot be one" because it violates sound logical reasoning. The problem, it seems to many, is that the claim that "one equals three" is simply absurd.
This difficulty occurs because of a failure to stipulate that the way in which God is one is not the same as the way in which he is three; the misstatement is something along the lines of, "God is three persons, but at the same time God is one person". However, the proper statement of the doctrine is that God is three distinct persons who share a single nature. Moreover, it is vital that we attach the proper meaning to the word "person" and the word "nature".
The short statement of the doctrine is, as we have heard it all our lives, that there are three persons in one nature. But if we attach no meaning to the word person, and no meaning to the word nature, then both the nouns have dropped out of our definition, and we are left of with the numbers three and one, and get by as best we can with these.
The doctrine may be set out in four statements:
- In the one divine nature, there are three persons - the Father, the Son and the Holy Spirit.
- The Father is not the Son, the Son is not the Holy Spirit, the Holy Spirit is not the Father: no one of the persons is the either of the others.
- The Father is God, the Son is God and the Holy Spirit is God.
- There are not three Gods, but one God.
We are not saying three persons in one person, or three natures in one nature; we are saying three persons in one nature. There is not even the appearance of an arithmetical problem. It is for us to see what person is and what nature is, and the to consider what meaning there can be in a nature totally possessed by three distinct persons.
Frank Sheed, Theology and Sanity (Ignatius Press, Second Ed., 1978), p. 90 ff
In theology the terms "nature", "essence", and "substance" are synonymous and refer to that which underlies all outward manifestation; the reality itself, whether material or immaterial. Unfortunately, in popular use the words have been diluted to something less than the total reality of a living thing; for example we say, "well, that's essentially correct" and we often mean it's partially or mostly correct; but the phrase properly means that it's correct in every way that matters.
Nature answers the question of what we are, whereas person answers the question of who we are. Every being has a nature, though not every being is a person; only rational beings are persons. Nature speaks to capabilities, limitations and will, where person speaks to self, emotions, intellect and passions.
This is a subject of much pondering and reflection; only God knows himself perfectly and we cannot be certain where, and even if, the division between what we are and who we are is made. But we can intuit a conceptual difference. We look inward and we identify a thing which we call "I" and see that it is distinct from that part of us we call "what". Even so, we are an integral creation; an amalgam of body and soul (or body, soul and spirit, depending on your interpretation of scripture) which, properly, wholly constitutes a human being.
Infinity and Eternity
In suitably contemplating ourselves, in seeing our "who" and our "what", and in seeing that in some measure our "what" is common to all humanity, we may begin to get a glimpse of what it might mean for a single nature to be shared by three distinct persons. In similar manner, we may begin to dimly comprehend what it might mean for one of those three persons to have two natures with respect to the incarnation. But in truth, it baffles the intellect, and in this life we shall only grasp it tenuously, incompletely. It will remain a mystery.
But of an infinite nature, we have no experience at all. If God tells us that His own infinite nature is totally possessed by three persons, we can have no grounds for doubting the statement, although we may find it almost immeasurably difficult to make any meaning of it. There is no difficulty in accepting it as true, given our own inexperience of what it is to have an infinite nature and God's statement on the subject; there is no difficulty, I say, in accepting it as true; the difficulty lies in seeing what it means.
Frank Sheed Theology and Sanity (Ignatius Press, Second Ed., 1978), p.95 (emphasis mine)
The doctrine of the Trinity solves a problem that purely unitarian conceptions of God cannot -- how can an infinite and eternal being experience love and personal relationship from eternity?
The only adequate object of infinite love is an infinite being, God himself. Certainly there is a real truth in the concept of God's loving himself infinitely, but it is not a truth we can make much of [...] we know that it is not an infinite egoism, but we cannot be rid of that feeling about it. But with the doctrine of the Blessed Trinity, that feeling vanishes: there is an otherness within the Godhead. Infinite love among three who are infinite with one same infinity means infinite love infinitely received, infinitely returned.
Frank Sheed, Theology and Sanity (Ignatius Press, Second Ed., 1978), p. 122