How can I develop leadership skills in graduate school?
Actually, I am surprised your graduate school doesn’t offer these kind of trainings. Graduate schools prepare PhD-students for their future life, whether in academia, industry or entrepreneurship. My graduate school offers for example also trainings on: leadership, group dynamics, teamwork, effective negotiation, autonomy and self management.

You probably have some room for ‘electives’ in your curriculum. Maybe you can use that room and follow your prefererred trainings in another (nearby) graduate school or in private sector. It may bring some additional costs and your department should be prepared to pay for it. 

I would certainly contact your graduate school about your needs because these are legitimate and in line with the objectives of a graduate school. In my university graduate students can also contact an HR advisor to talk about their future career development. I would be highly surprised if your graduate school or HR advisor could not / would not help you in shaping your future path. Maybe your request even helps your graduate school to improve or expand its curriculum.