Most companies (if not all) are required to submit form I-9
For clarification's sake, all employers, without exception, are required to obtain a completed I-9 form from every employee. There are a very few employees excepted; they include those employees who have worked for the same employer since prior to November 6, 1986, and employees who worked for the employer for less than 3 (three) days. We could get into a discussion of the definition of employee when it comes to varying forms of self-employment and corporate ownership but that's for another thread.
The I-9 requires that the employer or his authorized agent personally view the original documents provided by the employee. The employee must provide, within the first three days of employment, either a document that proves both identity and employment status (so-called List A documents; example, a passport) or one document that proves identity AND a document that proves employment eligibilty (so-called List B and C documents - examples, a drivers license (B- proves identity) and social security card (C - proves employment eligibilty) ). The lists appear on the back of the form; there are documents for both citizens and non-citizens that will satisfy the I-9.
It is entirely up to the employer whether to keep copies of the documents or not, and there are two schools of thought among HR professionals as to whether or not it is advisable. I personally belong to the Not school, but I acknowledge that the Keep school have some valid arguments on their side. Whichever the employer decides to do, though (keep or not keep), they must do whichever consistantly.
The forms themselves (with or without copies of the documents) must be kept on file until the later of one year after termination, or three years after hire. They need not be submitted to any government agency, but they must be available for view should ICE show up for an audit. Conventional wisdom is that I-9's should be filed in a separate file and not in the employee's personal file, but that is best practice, not law.