The "1000" years is figurative and represents the period of Jesus earthly physical presence through his return at the last day to gather up the saints. The Kingdom of God is not of this world (no, it does not spring from this world), but includes all of this world. All people and all governments are within the Kingdom of God as God's Kingdom rules the entire earth. Those governments that do not administer God's laws are in rebellion, which is every government to one degree or another.
The reality is that amillenialsts are really post millennial as they believe Jesus will return after the millennium, they just don't believe the millennium is a specific 1000-year period with Jesus physically ruling from a central office. But amillenialists need to recognize that God's Kingdom is more than a spiritual or Heavenly Kingdom as I described above.
As such, Christians need to be the salt of the earth and work to better society in whatever ways they are gifted in, so to be in government, education, industry etc, making moral decisions where others might be less than moral. If enough Christians had been doing this over the past 100 years or so, we might not be in the mess we are in now. But it seems the Church long ago gave the reins of secular society to the humanists.