Given the triune God, one nature and three persons, do they experience their will in their singular nature or in their separate persons? That is, do they share one will, or does each have a separate will which is in perfect harmony with the others?
While I'm not 100% certain I understand your question. If you're asking what I think; Jesus implied that the will of the Father, Son, and Spirit are separate, as He spoke a lot about the Father's will.
So I think Jesus clearly implied that He had the ability to disagree, but He chose to do the will of the Father. I think the clearest statement of this is in John 8:27-29, NIV
While the answer from WhatAboutJohn3_17 is fantastic, I thought I would at least add some more resources, to help give more depth.
I think the difficulty is trying to think of the Father, the Son or the Holy Spirit as existing separately from God.
This would be related to the question about the person of God, in relation to the Trinity, as explained by St. Thomas Aquinas:
God is a self-subsistant person that embodies all that is perfect, so though we label them separately, they are one.
To see if they have equality and what their likeness is like you can look at
but, basically, they are co-equal in all respects, so the Father is not over the Son.
Then you may want to look at the essence of the three persons:
Though it isn't spelled out in Scripture that God and the Son have the same essence Jesus did state "The Father and I and the Father are one (John 10:30)", for example.
So, the God is a person and has the same essence, and the person and property of God is the same (http://www.newadvent.org/summa/1040.htm), then, though this is a bit long-winded, but since God is a single person then we can see that though the Trinity may be separate, we can't differentiate between the will of each and the Will of God, as our sense of three does not force God to be that way (as explained more below). To see more about the Will of God you can read http://www.newadvent.org/summa/1019.htm.
You may want to read about the plurality of God (http://www.newadvent.org/summa/1030.htm) especially Article 3, where this comment is made:
So, the idea is that when we talk about three persons, that does not signify anything about God in reality. We have one God, and though, for our sake we express God as three persons, that is still for our simplicity, it does not mean that we have three gods that are one God, so, God will have one will from God, as, just because we think of God as three person doesn't mean that that changes what God is.
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".
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.
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?