I am a dual citizen of the US and another country. I have been using my other passport to identify myself to the TSA for the last few months (2 flights so far). The passport contains no visa, of course, nor entry stamp. They've never shown any indication of being interested in my immigration status. I hand it to them open to the picture/data page, and they never look at any other page.
I must add that I have never encountered the US Border Patrol at a TSA security check, but they do screen domestic passengers in some airports, including McAllen, TX. For example, see http://www.mercurynews.com/immigration/ci_26165754/what-does-border-patrol-do-at-airports-other.
Since the San Diego airport is less than 100 miles from the Mexican border, the Border Patrol is also able to screen domestic passengers there. A passenger flying in a domestic flight from San Diego ought to look into whether they actually do so.