In Matthew 25:31-46, Jesus gives us a picture of the Judgement. Notice that there are ONLY 2 groups. The ones on the right, His sheep, those that followed Him, will
inherit the kingdom prepared for you from the foundation of the
world (v. 34)
; and the ones on His left, the goats, who chose not to follow in His footsteps, will go
into everlasting fire, prepared for the devil and his angels (v. 41)
Notice that heaven was prepared for us, but the fire was prepared to the devil and his angels. Revelation 20 also gives us a closer look into Judgement.
It is the same, the people who have received the seal of God, whose names were written in the Book of Life, they will sit on thrones; while the devil, his angels, and those who were not found written in the Book of Life are cast into the lake of fire. If we follow the succession of Revelation 20, the devil and his angels are tossed in the lake of fire first and then the ones not found in the Book of Life. Then it says,
And death and hell were cast into the lake of fire. This is the second
death. And whosoever was not found written in the book of life was
cast into the lake of fire. (Rev. 20:14, 15)
Satan is cast into the lake of fire, the second death, eternal death and separation from God, because they did not wish to follow the Truth that was revealed to them.
There are many schools of thought on the matter. Based on the parables of Jesus it seems to me that the punishment, hell, will come AFTER the Judgement. I think this is only fair because it would be unfair to condemn someone to 10 years of prison without a fair trial and judgement. In the same way it would not be fair to send people to burn in hell and then be judged. This is a VAST subject and cannot be comprehensively discussed in detail in a single post.
In short, Satan does not punish anyone in hell, because no one has been judged and condemned.