Hosting company prepares an intern/trainee plan according to the instruction for DS-7002. The form can be downloaded at:

https://www.state.gov/documents/organization/84240.pdf

The placement plan must be sent to EEF. The intern must carry a copy of it, so that they can present it to an immigration officer, if required.
As Green Card or U.S. passport holders do not participate in the visa application procedure, which includes the insurance of the intern for the time of their stay in the U.S., it is important to remember about insuring the intern (which is then done by the EEF or the hosting company).
The Green Card or dual citizenship holders have a Social Security Number, so there is no need to apply for its issuance.