The FAA will not be checking on your immigration status.
The previous post is correct, you have to fill out a form and send it to the FAA in advance so that they can verify your foreign pilot certificate. This takes quite some time.
The form is available on
www.faa.gov
The Immigration service is now part of the department of homeland security, it's called the Bureau of Citizenship and Immigration now (BCIS) and the website is
www.ins.gov
You cannot take flight training legally if you are here on a tourist visa (or coming from australia - no visa is required for short stays).
If you are coming here to live then you are required to have your visa organised prior to arrival. Flight schools are not required to check on your immigration status, but rest assured, many of them do. Lots of states/counties are also requiring local flight schools to conduct background checks prior to giving flight training on light aircraft, although the FAA has issued an opinion saying that flight training does fall within the jurisdiction of local authorities.
So do your homework first. A good flight school will be able to help do a lot of the legwork for you.
If you want to read the regulations concerning conversion training go to
www.faa.gov also and see the section on Part 61 of the FAR's (federal aviation regulations).
Also this topic gets covered about once a month so search the archives.