All new employees must complete Section 1 of Form I-9 on (or before) their first day of employment. Then, within three business days following their start date, they must submit acceptable proof of their identity and eligibility to work in the United States.
As for your end, you must complete Section 2 of Form I-9 within those same three business days with the employee’s document(s). If the duration of the job will be fewer than three days, you must complete Section 2 no later than the first day of employment. Section 2 is generally done at the time the employee brings in their identifying documents, as it asks for specific information about these forms of identification. As a reminder, documents must be presented in-person, and the employer representative that reviews them must fully complete Section 2 of the form.
You can read about storage of Form I-9 (as well as other employee files) here.