Career change to economics from software engineering
If you are looking for a job with a title "economist," I doubt that anyone would look at you if you do not have at least an undergraduate degree in economics (in North America, anyway).

However, you could end up working for a firm that does economics, or financial research. I had a background in control systems engineering, with summer job experience as a programmer, and then ended up working as a quantitative analyst for fixed income. The programming experience ended up being probably the training that I ended up using the most. However, there is a lot of competition to get a job in finance.

Outside of finance, I believe that there is a lot of work in "big data," which involves a lot of programming.

You could end up working with economists, but you would probably not be considered an "economist."

You ask:


  Lastly, do economics ever serve in the private sector. I'm asking this
  because it seems that a lot of econ jobs would be as advisory roles in
  government institutions.


There's a lot of market economists in finance. A few large corporations still have people with the title "economist," but my guess is that most people with economics training have a different title when working elsewhere in the private sector. (As noted by denesp, Google famously has someone wih a Chief Economist title.) Public sector jobs are different in that respect.