Although I'm not sure if the Police have access to date of entry to USA and point of entry, it is certain some States have given their Police force access to immigration records.
Some states - post 9/11 - have elected to institute a Legal Presence proof for all citizens of the state to present when applying for State IDs, Driver Permits, and DLs. So, it is no surprise that the status of the individuals will be entered into the DMVs database and accessed by the Police.
As far as I know, the Police does not have any jurisdiction over immigration based crimes/issues. Meaning that if someone - for example - was arrested for possession of marijuana, the Police would charge them for possession under the State Law (a crime), and the INS would have jurisdiction to charge the individual for any Immigration crimes. By the way, if you are a greencard holder, you can lose your GC if charged with such a crime. But again, only INS can initiate that - not the local Police. In this case, there are 2 ways the INS will get notified of the crime - 1) the local Police forwards them the information or 2) the individual leaves the USA and comes back and the info comes up when the GC is swiped.
You have to keep in mind that the A-# and the SSN are now linked to the immigration/DMV/SSA databases in almost all states. So all records and convictions come up for INS.