I've not yet worked anywhere big enough to have a distinct Operations team sadly. We (development) always write the theme tune, sing the theme tune. This is now changing but slowly enough. I hope to do some of this learning that is spoken of and divest at least a few responsibilities!
I consider myself a good database Developer/Admin, but you wouldn't want me messing with your website configuration. Where I work now, we have a dedicated DevOps team. They do things like IIS configuration, domain account administration, instrumentation and automation for deployments, etc. Essentially they perform an assortment of roles and tasks across the enterprise that in some organizations a developer or database admin might have been expected to do occasionally, except a DevOps specialist takes it to the next level professionally. But, of course, everyone in IT is a link in the overall DevOps chain; it's not just a department. Even a small company with a handful of IT employees can apply the principles of DevOps.
This division of work also keeps developers from having unrestricted system access as both control over code and administrative access is often considered a poor situation when considering security.
If you use the right tools (we are using MS Release Manager), you can perform the deployments with a service ID, and not have any human intervention at all, other than having a suitable approver act as the final check on the move to production. We are automating almost every phase of our processes, from backups to acceptance/development from production to executing SQL scripts, to deploying code. I wish we had these tools a long time ago - it would have prevented a number of issues over the years.