How do Trinitarians interpret John 14:10?

In John 14:20, Jesus says his disciples will realize he is in them, and they in he. I don't think it would be appropriate to think the disciples were equal with Jesus' divine nature, whether understood by oneness theologians or trinitarians. The context is about Philip asking to see the Father, but Jesus, being the incarnation of God's word, exemplifies the Father through himself with the authority and will of the Father. The disciples will be able to do this as well once Jesus returns to the Father where the Holy Spirit will empower and guide the disciples.

The Orthodox Christian interpretation is that the essence of the Father and the Son is the same, but that they are still different persons. This is the Christian dogma of h*m*usia that was affirmed at the First and Second Ecumenical Councils in the 4th century, and is stated in the Nicene Creed. At the First Council of Nicea in 325 the Church affirmed:

We believe in one Lord Jesus Christ, the Son of God ... that is of the essence of the Father [ἐκ τῆς οὐσίας τοῦ Πατρός]
... being of one substance with the Father [ὁμοούσιον τῷ Πατρί]

Theophylact of Ohrid (1055-1107), summarized the Orthodox understanding of the passage you cite as follows:

The Son is in the Father. This means that the Son is revealed in the essence of the Father, and the Father is revealed in the essence of the Son. Similarly, a king is visible in his portrait [if it is well drawn], and the portrait is visible in his person. This is because of the characteristics shared by the portrait and the king. To show that His essence is the same as that of the Father, Christ declares, The words that I speak unto you I speak not of Myself. This means, "I speak only what the Father speaks. I have nothing of My own that is separate from the Father. Everything We have, We possess in common. We share one essence in two distinct hypotheses [persons]. The words I speak and the works I do are the Father's. Because Our works are divine, the Father is God and I am God: they are the works of a single essence. Whatever I do, the Father does. Whatever the Father does, I do.

Believe that I am in the Father, and the Father in Me

This means, "The very names Father and Son are sufficient to indicate that Our essence is identical. However, if this does not convince you that Our works are one in essence and honor, and that the Father is manifested in My essence and I in His, then believe Me for the very works' sake, for they are the works of God."

Explanation of the Gospel of John (tr. Chrysostom Press), pp.227-228

Perhaps, however, as you suggest some sort of argument supporting a different theology can be constructed on the basis of John 14:10 alone. This would be a very weak theology, though, would you agree? Any dogma proposed would have to find support throughout all of Scripture, without contradiction. We have for example, the testimony at the very beginning of John's Gospel (1:3):

In the beginning was the Word, and the Word was with God, and the Word was God. He was in the beginning with God; all things were made through him, and without him was not anything made that was made.

Furthermore, if the "Son" was only the "flesh", how do we interpret the tripartite division of body, soul and spirit? (1 Thessalonians 5:23). Did the "Son" refer only to the body, but the soul and spirit to God? How could the body only then pray to God the Father, as Jesus is shown to do in the Gospels.

All becomes clear when you consider the context, then link it with other things Jesus had already explained about his relationship with the Father. The context was the apostle Philip asking that they see the Father. Read from verse 8. This exasperated Jesus, for Philip had expressed his faith in Jesus as the promised Messiah the very first day they'd met, yet now, years later, he still didn't have experience of God as Father through all this time with the Son, "who is the express image of the Father". See John 1:45 & Col.2:2-10; The fulness of the Godhead dwelt in him bodily.

That is why Jesus rebuked Philip, saying, "Have I been so long time with you, and yet hast thou not known me, Philip? He that hath seen me hath seen the Father." (vs.9) To know Jesus is to know God as Father (vs.7). Then comes the verse you ask about. Matthew Henry's commentary on this is worth quoting.

"See here what it is which we are to believe (v. 10, 11)... as he had said (ch.10 vs.30) "I and my Father are one". He speaks of the Father and himself as two persons, and yet so one as never any two were or can be. In knowing Christ as God of God, light of light, very God of very God, begotten, not made, and as being of one substance with the Father, by whom all things were made, we know the Father, and in seeing him thus we see the Father. In Christ we behold more of the glory of God than Moses did at Mount Horeb.

...He spoke not of himself only, but the mind of God according to the eternal counsels... The Father is said to dwell in him ho en emoi meno~n - he abideth in me, by the inseparable union of the divine and human nature: never had God such a temple to dwell in on earth as the body of the Lord Jesus - ch. 2:21." (Commentary, p1608 first column, Matthew Henry)

The Trinity doctrine may be viewed as the Father and the Son sharing the one, divine nature, with absolute unity of the Spirit in that nature. That is how Trinitarians can explain John 14:10, for while the Father remained in heaven during the Son's incarnation on earth, the Holy Spirit was their constant link, as it were. Thus we see all three Persons of the Godhead manifesting themselves distinctly at Christ's baptism. The man, Jesus, came up out of the water, the Holy Spirit appeared above his head as a dove, and God spoke in approval of his Son from heaven.

No. It is as the apostle John stated at the outset of that gospel - the uncreated Word of God, who made everything that was made, was with God in the beginning, and was God. Logically, the one who made everything that was made could not have been made himself. He left his glory in heaven to add human nature to his divine nature. All three Persons in the Godhead have their distinct roles and functions, and one relates to the others in special ways, yet they share that one, divine nature. One God, three Persons, from before creation into eternity.