Not exactly...
We are in the '1,000 year' period of reign right now.
Satan is currently bound.
Satan will be released for a short period after the present '1,000 year' period....of which, the end will come because he has regained unbridled authority to possess the leaders of nations.
Yeah, that is not the truth at all. This is the claim of those who are amillennialis/preterists. The problem with this claim is that none of the characteristics of the millennial period exist. When Satan is bound, he is restricted to the literal location of the Abyss. This is the same place that those demonic beings collectively called Legion begged Jesus not send them into. It is also the same place that those demonic beings are released from at the fifth trumpet. The point of Satan's binding in the Abyss during that literal thousand years is so that he cannot deceive the inhabitants of the earth any longer and that because he will be under the earth in the Abyss and will have no exposure to people whatsoever.
Also, during the millennial period, scripture is clear that Jesus will literally be ruling here on earth from Jerusalem fulfilling the prophecy of ruling from the throne of king David. The millennial period is a time of peach with the Lord ruling with an iron scepter. Some of the characteristics of the millennium are that prey and predator animals and their young will lie down together and a young child will lead them. A young child will put his hand into a snakes den and not be hurt at all. If a man only lives to be a hundred years old, he will be thought to be accursed. Also, if a man dies at a hundred he will be thought to be a mere child. These are just a few of the characteristics of the millennium and they do not currently exist. When have you seen a lion eat straw like the Ox? That's another one.
The only that you can make these claims is by spiritualizing prophesy. Before Satan can be bound, the chronological order of events is that the world will have to first go through the wrath of God, which is the day of the Lord and then Jesus will return to end the age, then, at that time Satan will be bound. All one has to do is look at the state of the world to realize that Satan is not bound.