They definitely don't look. Several itineraries have multiple legs, and the TSA only ever looks at - much less scribbles on - your first leg. Even that aside, you can easily get a boarding pass from an agent inside the terminal, without it ever having been checked by the TSA.

I've actually gone through TSA with one boarding pass on one flight, and boarded a completely different flight before (not just a separate piece of paper) - back when I could book flights for free on JetBlue and had already booked another flight that night. I merely decided once I was in the terminal that I'd hop on a different flight I had also checked in to.

I do a lot of flying and have long though about this. It's total theatre. They could fix it by implementing some cryptographic code that's scanned at TSA entry points, verifying the actual document (boarding passes are a far cry from a verifyable document).