In the doctrine portion of Lecture Two, one of Edwards' points is that though the "extraordinary gifts" (what we would call spiritual gifts or charismata, particularly prophecy, tongues, miracles and the like) are great privileges, the ordinary influence of the Spirit (the fruit of the spirit, which are summed up in love) is far more excellent. In explaining the reasons for this superiority, he made a point which some of us found difficult to wrap our minds around and which led to a fairly lengthy discussion:
Unlike extraordinary gifts, the "This blessing of the saving grace of God is a quality inherent in the nature of him that is the subject of it." Rather, he says, "They are like a beautiful garment, which does not alter the nature of the man that wears it. They are like precious jewels, with which the body may be adorned; but true grace is that whereby the very soul itself becomes as it were a precious jewel."
I think our discussion ended with good comprehension, but since that evening I've had occasion to revisit Edwards' later work, Religious Affections where he addresses this very subject in much more depth. So I will let him speak for himself and see if he can't clear up any remaining confusion anyone may have.
"The Spirit of God is given to the true saints to dwell in them, as his proper and lasting abode; and to influence their hearts, as a principle of new nature, or as a divine supernatural spring of life and action. The scriptures represent the Holy Spirit,not only as moving, and occasionally influencing the saints, but as dwelling in them as his temple, his proper abode, and everlasting dwelling place. (1 Cor. 3:16; 2 Cor. 6:16; John 14; 16-17). And he is represented as being there so united to the faculties of the soul that he becomes there a principle or spring of new nature and life.....
On the other hand, though the Spirit of God may many ways influence natural men; yet because it is not thus communicated to them, as an indwelling principle, they don't derive any denomination or character from it; for there being no union it is not their own. The light may shine upon a body [ie. the moon] that is very dark or black; and though that body be the subject of the light, yet, because the light becomes no principle of light in it, so as to cause the body to shine, hence that body don't properly receive its denomination from it, so as to be called a lightsome body [ie. the sun]....
The Spirit of God so dwells in the hearts of the saint, that he there, as a seed or spring of life, exerts and communicates himself, in this his sweet and divine nature, making the soul a partaker of God's beauty and Christ's joy, so that the saint has truly fellowship with the Father, and with his Son Jesus Christ, in thus having the communion or participation of the Holy Ghost.
On the contrary, as regards extraordinary gifts, of which Christ tells us in Mt. 7:21-23, many will exhibit without being known of God or having experienced His saving grace, Edwards has this to say:
"But the Spirit of God never influences the minds of natural men after this manner. Though he may influence them many ways, yet he never, in any of his influences, communicates himself to them in his own proper nature. Indeed he never acts disagreeably to his nature, either on the minds of saints or sinners; but the Spirit of God may act upon men agreeably to his own nature, and not exert his proper nature in the acts and exercises of their minds [think of the prophecy of Caiphas in John 11: 49-53]: the Spirit of God may act so, that his actions may be agreeable to his nature, and yet may not at all communicate himself in his proper nature, in the effect of that action. Thus, for instance, the Spirit of God moved upon the face of the waters, and there was nothing disagreeable to his nature in that action; but yet he did not at all communicate himself in that action, there was nothing of the proper nature of the Holy Spirit in that motion of the waters. And so he may act upon the minds of men many ways, and not communicate himself anymore than when he acts on inanimate things....Ungodly men, not only han't so much of the divine nature as the saints, but they are not partakers of it; which implies that they have nothing of it; for a being partaker of the divine nature is spoken of as the peculiar privilege of the true saints. (2 Pe. 1:4). Ungodly men are not partakers of God's holiness (Heb. 12:10)." [all emphasis mine]
In short, the Spirit of God does not conform any into the image of Christ who are not His.