Having driven "non-CDL required" tow trucks in VA, PA, MD, DC, and not having a CDL, I have passed some check staions in VA with no consequence and have been stopped for passing a check station in MD. I guess it depends on the state and the check station. The only requirement I had was to have the CDL medical certificate with me. I have kept a log book (I was commercial) on longer trips but have never been asked to show it.