They have become callous and have given themselves up to sensuality, greedy to practice every kind of impurity. (ESV)
It has been said that, "Without God everything is permissible." This axiom really sums up the way that godless Gentiles live their lives. God is a holy God. He has established boundaries and guidelines for our behavior that enable us to live lives that do not harm one another. But, the Apostle Paul describes for us those who live outside of God's holy boundaries. "They have become callous and have given themselves up to sensuality, greedy to practice every kind of impurity."
This callousness is a description of gradually becoming harder and harder due to repeated behavior. I remember learning how to play guitar. At first it hurt my fingers to push down the strings to form guitar chords. But, the more I practiced I developed callouses on my fingers. The skin on the tips of my fingers hardened and I no longer felt discomfort when playing. This applies directly to those who repeatedly sin. The more that they engage in their wickedness the easier it is to continue on in it. They no longer feel any unpleasant feelings of a conscience. This frees them to chase every type of sin which Scripture reminds us leads to their eternal death (Romans 6:23).
Now, I find a very interesting phrase in this verse in Ephesians. Paul says that they have given themselves up to sensuality. In Romans 1:28 Paul was describing Gentiles also. He talks about those who have received the knowledge of God, but rejected it in order to pursue pleasures. Listen to how he describes them:
 And since they did not see fit to acknowledge God, God gave them up to a debased mind to do what ought not to be done. (ESV)
In Ephesians he says, "they have given themselves up." In Romans he says, "God gave them up." Is this a conflict? Actually, no. God gave them up to do what they gave themselves up to do. Here is the scary truth. If you want to continue on and pursue your rebellion, God will let you. Be careful what you ask for, you just might get it!
Don't be surprised when your friends, co-workers, and family members pursue sinful behaviors without having any sense of conscience. Paul describes them as those who have hardened themselves in their pursuit of sin. Spend time praying that God will soften them and draw them to himself. Pray that they do not ultimately get what they ask for!