Do any Christian groups sell all their possessions as it seems the Early Church did? In Luke 12:33 (NIV) Jesus says:


  Sell your possessions and give to the poor. Provide purses for yourselves that will not wear out, a treasure in heaven that will never fail, where no thief comes near and no moth destroys.


He doesn't say "all" possessions here, but that seems to be what he told the rich ruler in Luke 18:22 (NIV):


  When Jesus heard this, he said to him, "You still lack one thing. Sell everything you have and give to the poor, and you will have treasure in heaven. Then come, follow me."


In Acts 4:32-35 (NIV) it looks like that's what the Early Church did as well:


  All the believers were one in heart and mind. No one claimed that any of their possessions was their own, but they shared everything they had. With great power the apostles continued to testify to the resurrection of the Lord Jesus. And God's grace was so powerfully at work in them all that there were no needy persons among them. For from time to time those who owned land or houses sold them, brought the money from the sales and put it at the apostles' feet, and it was distributed to anyone who had need.


Are there any denominations or other groups in the world where the congregation (not monks, nuns etc.) actually does this? If so, can you please give some details on how the practical aspects work (sharing homes etc.).
I don't know if there are any denominational-scale organizations that practice this, but there are certainly churches and other Christian organizations that practice this.

The Bruderhof may be a good example of this. My family lived in a Bruderhof community for a few months when I was growing up, as my father was doing some work (writing a book) for them.

Later, a Mennonite church I used to attend had previously lived like this, in community, requiring all members to donate all of their earnings to the church, which in turn provided the bare essentials for living (home, groceries, etc), and gave the rest away.

They eventually gave up that practice for legal and tax reasons, and reverted to individual ownership, but with a spirit of giving.

In both cases, the church/community owned shared property, and was responsible for providing for the needs of the church members. In the case of the Bruderhof, the church also owned land for growing crops, a furniture factory, and perhaps other businesses, which were then used to sell goods to the general public, and provide for the needs of the community.

I think it won't be hard to find examples of Christian communes living in a spirit of "complete giving" in various parts of the world.

A google search for "Intentional Christian Community" finds various resources on this type of living, and may help you further in your search.