Well, if they're not here legally, they go to jail anyway. We don't write them a ticket based on the information they give us and trust that's who they are and that they'll take care of the situation. It goes like this - I stop an illegal alien. I ask for a DL and he says "I no have one" or "I no speak engless". I have him give me his name, d.o.b., etc. I return to my trusty computer in my car and put the info in that he gave me. If I can't find positive identification on him, he goes to jail. Once in jail, he can have someone bring a valid (read: US issued) form of identification, he can post a bond, or he can wait to see a judge.
Anytime an illegal immigrant is arrested, the jail contacts ICE who usually refuses to come pick up the individual. ICE's stance on the issue changes quite often. Occasionally, they'll come pick up someone who was arrested for a simple traffic violation, but 95% of the time, they won't come get them unless they were picked up on a felony. It's also important to note that they won't deport anyone until all charges are taken care of. ICE will put a hold on the individual and (usually) when the judge sees the ICE hold, they'll dismiss the charges (or sentence them to time served) so that ICE can come and get them for deportation.