It doesn't have to be a full amnesty, but whatever.
What your side keeps overlooking on this issue is the nuts and bolts of how you accomplish this on the ground level. The biggest single problem is being able to identify these people, and their immigration status, quickly and easily. And without the incredible ease of forgery in the current system.
Right now, all you basically have to do is show your social security card. How easily forged is that? Contrast that to your credit card. People may steal them, but forging them is incredibly difficult because you've got to fool the computer reading the unique I.D. number, not the schlub clerk who rings you up.
The absolute key to any effective enforcement plan is getting a quick, accurate, and secure I.D. system. Part of which is you've gotta get them to come out of the woodwork to sign up for it. Thumbprint verification, etc. Do that, and the ability to control who actually gets to work here and/or stay here skyrockets. That's what was missing completely from IRCA in 1986.
Link those cards to receipt of tax payments by the IRS. "Guest worker" doesn't actually work and have withholdings taken out by employer, and his card is flagged for deportation. Time in country expires, card is flagged. And any employer/INS agent can verify that in 60 seconds. You can put whatever time limits on stays you want, and for the first time, it will be subject to real-time verification.