I may be wrong, but I’m not sure if the statement that “a person is a mind” necessarily is true. This is certainly the case for humans, but human personhood is at best analogous to divine personhood. To answer your question, each of the three persons possess the single divine nature in its fullness. I think it helps if we consider the relationships between the three persons. The Second Person is the result of the First Person perfectly knowing Himself (since the knowing is infinite, the known Person is equal in every way to the Knower). The Third Person is the result of the infinite pouring of the First and Second Person into the love of each other. So it may be more apt to say that the single, metaphysically simple nature of God is expressed by the three persons: the infinite lover, the infinitely beloved, and the infinite spirit of love between lover and beloved. Does that help answer your question?
I sort of agree and sort of have a problem with what you say, but no matter what any human being can say, I/we would probably have a problem. At first I said, balto nailed it, then almost immediately I said, wait a minute, the Father is not the only one who loves, the Son loves the Father, so in that way the Son becomes the knower and the Father becomes the known, and no sooner did I see that then it occurred to me that the Holy Spirit is no laggard or passive part with respect to love, so to speak, he too loves the Father and he too loves the Son, because after all, he is more than a “product” of the Father and the Son’s love, he himself is a Person and wholly God, SO then it occurs to me that it all goes back to the idea of begetting, begotten and proceeding, but who can fathom these seemingly time-related words as an infinite reality. So I said to myself, you are beginning to chase your mind’s tail. For next you will be saying what is the love between the Holy Spirit and the Son called, and likewise between the Holy Spirit and the Father? And if the idea of begetting, begotten and proceeding will not allow it to go that way in terms of producing another Person, we must still recognize that the Holy Spirit really does love both the Father and the Son, since he is in every way God, but for reasons unknown this does not, at least to my knowledge, result is another Person, or the Father or the Son (because of that mysterious concept of begetting, begotten and proceeding, I think). Now I almost have myself confused, so I am sure I lost you, but it seems to me that trying to pursue the idea of reconciling the Trinity would take some time, in fact, and infinite amount of time, which itself is an absurd idea, I think, for infinity resides outside of time.

So, I think for now I’ll just remember the formulae, there is but one God in three Divine Persons, Father, Son and Holy Spirit, each of whom are fully God.
I would like to know why the Catechism says, “He is not the Father who is the Son, nor is the Son he who is the Father, nor is the Holy Spirit he who is the Father or the Son.” Why doesn’t it read, “He is not the Father who is the Son, and he is not the Father who is the Holy Spirit, nor is the Son he who is the Father, nor is the Holy Spirit he who is not the Father?” Is it heresy to say it this way? Why the insistence on always “packaging” the Father and the Son, as if to parallel the begetter and the begotten, even when we’re not speaking in terms of their relationship, but only in terms of their equality? Is it a necessary construct even when not talking of their relationships with each other? Why do we never say the Holy Spirit loves the Father and the Son? He does, of course, but why do we never say it? I’m not meaning to be a troublemaker, only expressing the fact that these things trouble me.
I can’t even understand the simplest things the Church teaches about the Trinity. The furthest my mind can get with it, and I may be wrong, is “begets” is a short form for “infinitely begets”, and “begotten” is a short form for “infinitely begotten” and “proceeds” is a short form for “infinitely proceeds”, so that the modifier serves to strip the verbs of their relationship to time in normal usage. I suppose these things are taken for granted, but I am no theologian.
Just musings. No reply is necessary, but all are welcome.