A reader has sent along a helpful tip on a site (run by NumbersUSA, which opposes the "grand bargain"), SmartBusinessPractices, which has an outline of the way the current "Basic Pilot Program" works. This program is currently voluntary under federal standards, but all businesses, it seems, are welcome to use it. And participation in this program is now mandated by law in Arizona.
According to this outline, there are three basic "steps" in the Program's process.
Step 1: After registering with the Program, an employer has a new hire fill out an I-9 form (e.a.):

The I-9 simply states the employee's name, date of birth, social security number, and an attestation that the employee is a United States citizen, lawful permanent resident, or alien authorized to work in the U.S. The employee presents to the employer either one document establishing both identity and work authorization (e.g., a U.S. passport or green card) or two documents that together establish identity and work authorization (e.g., a driver's license and a social security card). The employer examines them to make sure they reasonably appear to be valid (i.e., the photo is not taped on the driver's license) and records the document number, issuing agency and expiration date, if any, on the I-9. Then, the employer signs an attestation on the I-9 saying that he has examined the documents and they appear valid.

For steps 2 and 3:

Step 2 - Once the I-9 is done, the employer logs onto a secure DHS website, enters the employee's full name, date of birth and social security number. He chooses from a drop-down box which document/combination of documents the employee presented, and then clicks the submit button. The information about the employee is transmitted immediately to the SSA (Social Security Administration).

Step 3 - If the SSN (social security number) and the name match SSA records, the employer receives a message within two or three seconds that the employee is authorized to work and the process is finished.

If the SSN and name match, but the SSA cannot verify that the employee is work authorized (i.e., the SSN may have been issued "not for employment purposes") the employer gets a message that DHS is attempting to verify work authorization. DHS usually responds within 24 hours, but the law gives it three days, since it has to check its records by hand if the automated check does not match the name and immigration document. If DHS finds a match, it tells the employer and the process is finished. Otherwise, the employer is told to have the employee check with DHS directly to clear up the problem.

To Hire or Not to Hire - If the SSN and name do not match, the employer receives a message to refer the employee to SSA to clear up the problem.

In either case where the employee is referred to SSA/DHS, the employer will be notified within 10 days that either work authorization is confirmed or not confirmed, in which case the employer must terminate the employee.