Data Science Companies
Some of the biggest companies in the world are 'data science' companies; facebook, for instance, performs targeted advertising based on analyzing users preference.  In essence, it's the most highly trained recommendation engine ever, as it knows a great deal about it's consumers.  At it's heart, Google is an advertisement firm, and similarly leverages user information to provide targeted messaging to consumers.  The same can be said for Amazon, Twitter, and most likely pokemon go.

If you're thinking of a more purely 'data science as a service' company, larger consultancy firms such as the Boston Consulting Group have always incorporated analytics in their services, and have certainly profited from the proliferation of data.  They don't say "we do data science", because people care more about getting the right answer, than how the answer is obtained.  

As far as Kaggle goes, while it's an awesome resource, and it provides cool tutorials and data sets, if you're a professional data scientist, then you don't work for free ;-)  If a billion dollar company can harvest even a 1% gain in efficiency by offering a $50K prize, they can recoup that money very quickly.  If you're only asset is intellectual property, you need to be cautious about giving it away.