Do the deal teams in private equity ever act as operating partners also?

I am just starting to learn about how the private equity industry functions, so bear with my ignorance.

From the research I have gathered it seems that there are two different types of private equity funds, you can either be an independent sponsor or the type that raises, manages, and deploys capital. These firms have deal teams and operating partners (or sometimes consultants or advisor).
I want to break into private equity eventually, but am drawn to the industry because the functions of both the deal team and operating team are very appealing. Are there firms (independent to large firms) that allow associates and up to operate on both sides of the fence? Do independent sponsors act in deal sourcing and operations?

Thanks in advance.