My [fill in the blank] Handles the Website
I regularly hear publishers talk about how so-and-so in the company handles the website, even when there's just two people in their company I hear "my wife handles the site".
Should I read this to then mean that you handle the magazine? Of course not, the person involved in driving the website spends the majority of their time - as they should - with their various publication-related activities.
OK, so what's my point?
My point is that while you surely should have a point person for digital/website stuff the way you have an editor or sales manager if you're involved with the business you need to be involved and understand the digital side of things.
For most there's not much to be involved with but for the "new media company" digital is part of the brand, part of the story, and part of the experience. Readers engage online and advertiser pay to be a part of the digital experience for the same reason they engage in your print-related activities. Digital and print should not and cannot be distinct but rather two different parts of your community-oriented brand and this starts with how your organization approaches things.
The best organizations do this seamlessly. You might wonder why it looks like they don't have to work as hard - well, it's because they don't. When you have a Media Mindset you can get more done with less effort because everything you do is taking you in the same direction - toward delivering a consistently compelling cross-media community engaging experience.