There's a big unanswered question. Is it random?
If any given search term is equally likely to be hidden, then you've still got the same information. Just less of it.
I think the information is likely to be the same until something that we're not yet seeing changes the nature of the hidden searches... ie, how something in the searching situation changes how searchers behave.
The largest such changes, I'm thinking, are likely to be in mobile, which is the area of largest anticipated growth. Mobile searches are secure and won't return referrers. Mobile interfaces are rapidly changing, and this might also affect the nature of searches.
Personalization conceivably will also change searching patterns by changing the results that searchers have seen before new searches.
The largest personalization factor I've thus far seen, btw, even on desktop, is location. More often than not, location is not entered into queries when localized results are desired.
On mobile, it's likely that results, which are in many cases already hyper-localized, are likely to become ever more narrowed down by place, and conceivably by time... and by intention as perceived by time and place. Eg, a mobile search from a downtown area at dinner time for a restaurant is likely to narrow down the geo-area quite a bit, with no location information entered in manually.
Conceivably, Google... or sites returned by Google... might even go into a restaurant app-like mod and offer a menu-driven dialogue to rapidly narrow down choices. Assuming that what we call "websites" receive any of that traffic, it's hard to say what the stats might look like.
I can envision similar localization for some types of shopping.