It's unlikely that companies would look to outsource such a critical component and also it would be near impossible to create trust around 3rd parties accessing their data especially via an intermediary service.

There are three major challenges with the scenario you describe:
1) the data is usually not in a format that is ready to be chewed upon (less critical)
2) The data-science tasks are usually not well-defined by the people who need them (very critical)
3) The process tends to be iterative and not on-shot.

The only successful situations I'm aware of that is close to the one you describe is competitions/benchmarks where the task is crystal-clear and the data is ready-made (like the Netflix one or many others run in the research community and by the government).

In these competitions, issues (1) and (2) are addressed, and you may hire a person so that they can iterate on it and continue the work (issue 3).

In certain domains and projects there's less iteration needed, so if the problem to be solved is well-defined and data is well-prepared, it can be done successfully. I've been in several situations where I "ordered" a data-driven algorithm and plugged it in a live system.

I would copy Kaggle initially (80%) and innovate (20%). The key is to get to Product / Market Fit asap, then scale.
http://www.kaggle.com/

Andrew Chen ( https://clarity.fm/andrewchen ) goes over this nicely in his presentation Zero to Product Market / Fit
http://andrewchen.co/2013/10/14/zero-to-productmarket-fit-presentation/

So typically it would depend on the complexity of the project and how easily accessible is the data to determine the likelihood and possibility of outsourced data analysis.

For example, it is much easier to look at a ecommerce website and provide analysis than to go into a retail shop that has both online and offline data sources that needs to be analyzed.

As an web analyst, I tend to get projects to help identify opportunities and support marketing activities. There is limitations due to the amount of data available and it would usually need to be looked at on a case to case basis.

Big data can include data multiple departments (sales, accounting, operations) to something as simple as web analytics. For complex projects, you face issues such as security, accessibility, longer timelines and iterations which would probably mean an inhouse resources would make more sense.

Hope this helps. Be happy to hop on a call to address any additional issues.