How do you build an “ownership mindset” on your team? Is it possible to train people to think like this, or is it an innate quality you have to hire for?

Incorporating cultural, structural, and individual development tactics into a team's ownership mentality building process is complex and multi-faceted. This kind of thinking may be fostered in a team, albeit some members may be more open to it than others.

I have seen the implementation of the Objectives Key and Results methodology work well in at least two different digital transformation journeys. It connects the leadership team's vision to each employee's daily tasks. In conjunction with regular town halls, reinforcement from the managers and autonomy results in an "ownership mindset". When an employee understands how it contributes to the company's progress and suggest ideas, they naturally tend to own them. 

I agree with many of the comments here.

I would add that when talking about ‘building ownership’ we are often saying how do we make employees ‘act like an owner’ even when they are not.

Ultimately, if you want a true ownership mindset, people literally need to be made owners (ie partners / have a shareholding / participation in profits that is significant to them).

And that is of course that is complex for an org to manage. I don’t have a silver bullet. My point is building ‘ownership mindset’ through culture, autonomy, training, etc can only get you so far as to ‘acting like an owner’.

Over time you can absolutely develop a culture where employees embrace ownership, but with ownership comes accountability.  This means your organization must also support autonomy and risk taking, and a growth mindset that embraces a bias to action and a tolerance for failure.  Incentives must also be aligned.  Without some of these key pieces, your efforts to get people to 'own' things will likely struggle.  

We instill the concepts from Carol Dweck’s book Growth Mindset with our people and hold regular webinars to reenforce and keep it top of mind. We also have, within the Information Technology Office, a Cost Optimization program now in its third year and both of these contribute to the Ownership Mindset. The latter giving people purpose, and I would characterize as a journey. Is it possible to instill (train) these for attributes? Yes, but it takes time for people to rally and see the result. Once they see the benefits and realize they’re empowered to make decisions or bring things forward (this is more for cost optimization) then you make believers out of them. I don’t think we would have success with our Cost Optimization program, without the Growth or Ownership mindset.