My job, just like it was when I was a sysadmin, is to keep things running smoothly. To optimize both sides of the coin and to take things off their plate that I myself can deal with that might be a blocker to them finishing tasks and to us getting features out or server migrations done.
Some devs know ops and some ops know dev, those guys, more often than not I just check in with and see that things are running smoothly. For the other folk I try to create systems and workflows that get blockers out of their way.
I try to be a useful and reliable tool to both sides as much as I can be.
Hopefully the way I've been doing this for the last two years since I moved into a more Devops related role has been helpful to everyone.