There are a number of points amillennialists disagree on: the nature of the millennium, who takes part in the 'first resurrection', what the 'first resurrection is, etc.
But the main point of disagreement is when the thousand years of Revelation 20 begin:
1. Millennium begins at the cross (c. 30 AD)
Perhaps the more well-known variation of amillennialism associates the binding of the dragon with the crucifixion (and/or resurrection) of Jesus. Appeal is made to passages like John 12.31 or John 16.11 in combination with Matthew 12.22-29. Within this framework, the binding of the dragon is a symbolic reference to the satan suffering 'judgment' and being 'bound up' through the cross. Hence, the thousand years began at that time (i.e. about 30 AD).
2. Millennium begins at the destruction of the temple (70 AD)
Some amillennialists believe the dragon was bound when Jesus was crucified, but when the temple of Jerusalem was destroyed in 70 AD, and thus the millennium began then. This view is most common within a specifically preterist eschatology.
3. Millennium begins at the fall of Rome (4th-5th centuries AD)
And a third set of amillennialists see Revelation 19-20 as foretelling the collapse of the Roman Empire, whether by conversion (i.e. Constantine's reforms) or by conquest (barbarian invasion of Rome). Interpreting one (or both) of these as the triumph of Christianity over paganism, the dragon is believed to have been bound, and the millennium begun, around this time.
There is no direct conflict between the amillennialism expressed in views 2 and 3 with the activity of the satan as described in 1 Peter 5.8. Because, in these two views, the millennium begins after 1 Peter 5.8 was written, c.62-64 AD, it may be understood as only applying to the time frame between the epistle's authorship and the start of the millennium. (A critical dating of 1 Peter's authorship, c.80-100 AD, is more problematic for view 2.)
This leaves view 1 unaccounted for, the view that says the millennium began when Jesus died.
But reconciliation is made between 1 Peter 5.8 and view 1 (and really all three views) on the basis of an alternate understanding of what Revelation 20 is actually describing. The point of debate often centers around when the dragon is bound or when the millennium begins, but some authors have instead framed Revelation 20.1-6 around the nature of the dragon's imprisonment.
In specific response to the use of 1 Peter 5.8 as an objection to amillennialism, Riddlebarger argues:
The apostle John told us in Revelation 20 what this binding of Satan actually entails. Confined to the abyss, the devil is no longer free to deceive the nations until he is released before the end (20:3). This does not mean that the binding of Satan prevents all forms of satanic activity in this age.1
In his commentary on Revelation 20, G.K. Beale (who interprets the millennium as spanning 'from Christ's resurrection until his final parousia', i.e. view 1), also responds directly to the use of 1 Peter 5.8, saying:
But the "binding" (δέω) of Satan in Mark 3:27 (= Matt. 12:29) does not restrict all his activities but highlights the fact that Jesus is sovereign over him and his demonic forces. Therefore, context, and not the metaphor by itself, must determine what degree of restriction is intended. That Satan is "cast out" (ἐκβάλλω) by Christ's death does not restrict Satan in every way. Rather, it keeps him from preventing "all people" throughout the earth being drawn to Jesus (John 12:31-32). [...] God's seal on Satan prevents him from harming the salvific security of the true church, though he can harm it physically.2
In other words, amillennialists see the dragon's imprisonment in Revelation 20 not as an absolute restriction of all of the satan's activity in the world, but a more specific restriction of the satan's ability. G.K. Beale continues by defining what he thinks this specific restriction is:
But what kind of deceiving activities are to be restrained? The following context, 20:7-10, provides the nearest answer, especially since v 7 begins where v 3 leaves off (cf. "until the thousand years should be completed . . . it is necessary that he be loosed" and "when the thousand years are completed, he will be loosed"). In vv 7-10 the devil is released "to deceive the nations . . . in order to gather them together" to exterminate the community of God's people on earth. This will occur at the end of history immediately before Christ's final coming, when he will destroy the nations with fire.3
This is similarly argued by David Chilton (who followed a synthesis of views 1 and 2):
For all these reasons, it is generally suggested by both postmillennial and amillennial authors that the binding of Satan, so that he should not deceive the nations any longer, refers to his inability to prevent the message of the Gospel from achieving success. And, as far as it goes, this interpretation certainly has Biblical warrant: Before the coming of Christ, Satan controlled the nations; but now his death-grip has been shattered by the Gospel, as the good news of the Kingdom has spread throughout the world. [...] That Satan has been bound does not mean that all his activity has ceased. The New Testament tells us specifically that the demons have been disarmed and bound (Col. 2:15; 2 Pet. 2:4; Jude 6) – yet they are still active. It is just that their activity is restricted. And, as the Gospel progresses throughout the world, their activity will become even more limited. Satan is unable to prevent the victory of Christ’s Kingdom. [...] the precise thrust of Revelation 20 seems to be dealing with something much more specific than a general binding and defeat of Satan. St. John tells us that the Dragon is bound with reference to his ability to deceive the nations – in particular, as we learn from verse 8, the Dragon’s power "to deceive the nations . . . to gather them together for the war." The stated goal of the Dragon’s deception is to entice the nations to join forces against Christ for the final, all-out war at the end of history. [...] The specific point of the binding of the Dragon, therefore, is to prevent him from inciting the eschatological "war to end all wars," the final battle – until God is ready.4
In other words, amillennialists identify a difference between the activity of the satan in 1 Peter 5.8, and the activity the dragon is prevented from doing in Revelation 20. In the former the activity the satan does is understood as a more personal type of persecution, while the latter is understood as the dragon being restricted from carrying out an overt attempt at a global annihilation of all of God's followers. Amillennialists believe that a restriction of the latter does not include a restriction of the former.
1 Kim Riddlebarger, A Case for Amillennialism, p.242
2 G.K. Beale, The Book of Revelation, p.985-986
4 David Chilton, Days of Vengeance, p.199-201 (bold and italics original).