How do you prove you haven't had sex?
You can't, so you sign an affidavit saying so.
Most people can't afford to pay for two households so it's common to be separated in the same house. You sign saying you haven't slept together since whenever and then incorporate that into the MSA.
Illinois has a 2 year separation for irreconcilable differences, so what. I filed in September of 2009 and then this (a part of the MSA) was signed in August of 2010:
The parties live separate and apart from each other and have so lived separate and apart since March of 2009 and have executed a proper stipulation waiving the statutorily mandated two year period of separation for a Judgment of Irreconcilable Differences;