In the US, employers are required to implement a procedure called I-9 Employment Verification for each employee they hire. New employees must present proof of identity and proof of work authorization. The required standard of proof is specified by US Citizenship and Immigration Services (USCIS). Employees may either present one document that satisfies both requirements (list A), or two documents, one that covers each requirement (list B and list C). Most US citizens will either present a US passport book (list A, identity and work authorization), or a driver's license (list B, identity) and a birth certificate or social security card (list C, work authorization). Non-citizens with a valid work authorization will most commonly present a green card (permanent resident card) or a foreign passport with a valid work visa in it. Other documents are acceptable but less commonly used.
The employer is required to inspect each document and verify that it "reasonably appears to be genuine and relates to [the] employee presenting it," as well as fill out some paperwork (form I-9). This is then filed with USCIS, which may choose to investigate (and/or notify ICE) if the form contradicts their records. At no point are employers given access to US immigration data (however, employers participating in E-Verify can get a confirmation from DHS that the information submitted on the I-9 form appears to match those records). Most employers would not want to have access to such data anyway, because it is a liability. Employment discrimination on the basis of national origin is banned under the Civil Rights Act, and even having access to the data can make a discrimination complaint more difficult to defend.