In Illinois it actually is illegal to make audio recordings of on-duty cops--or any other public official. Illinois is one of a handful of states that require all parties to consent before someone can record a conversation. But the other all-party-consent states also include a provision in their statutes stating that for there to be a violation of the law the nonconsenting party must have a reasonable expectation of privacy. On-duty police officers in public spaces have no such expectation.
Here's where it gets even worse: Originally, the Illinois eavesdropping law did also include a similar expectation of privacy provision. But the legislature stripped that provision out in 1994, and they did so in response to an incident in which a citizen recorded his interaction with two on-duty police officers. In other words, the Illinois legislature specifically intended to make it a Class I felony, punishable by up to 15 years in prison, to make an audio recording of an on-duty police officer without his permission.