No, "public" and "private" just won't do.
How about this: the "productive sector" and the "nonproductive sector"? Admittedly, the "nonproductive sector" is a little cumbersome, so we may want to consider variants in wording, but you get the idea. Some would argue that the government and its agents, public-private partnerships (yuck!) and the like, occasionally produce goods and services people want, to which I would argue, "not without forcibly creating monopolies for themselves." Be that as it may, it is prudent to search for better, more universally accepted alternatives.
I don’t mean that a business politician won’t steal; stealing is his business. But all politicians are nonproductive. The only commodity any politician has to offer is jawbone. -- Lazarus Long