Since it's 2016, hopefully your company has developed some strategy to utilize social media, even if it's fairly limited at this point.
If you're still new to social media and/or you're still not seeing real results, there is a question you need to consider:
"Who should run your social media and be your company social media manager?"
There are a couple of different approaches to this question. Let's tackle them quickly.
Which department should Social Media Marketing reside in?
This depends on how exactly your company is structured. Most companies put social media under marketing, although some place it under 'consumer' or 'customer' departments.
If your company is larger and has a specific communications department, sometimes it resides there as well. Again, it varies, but it usually resides in a department with outward, client-or-customer-facing potential.
If marketing is the department that produces content for your business, then it probably makes the most sense to have social media also reside in that area. After all, once the content is produced ... social media is going to be a channel for you to push it out to a potential audience. Cross-functional collaboration (i.e. across different departments) is often very hard - different departments tend to have different metrics and tasks, and their leaders are accountable to different business aspects.
Getting ROI from social media is difficult enough as is is, don't make it harder by having social reside in a different department than content production.
Align those two areas. It will make the job of a social media manager much easier.
A few other thoughts here:
Everything needs to begin with goals. Once you understand your goals in social, it'll become more apparent as to where the responsibility should reside. Your social efforts need to tie to your broader business goals in order to have the most impact. A typical example is "acquisition of new customers" - you can acquire customers on social, and that's directly tied to revenue.
Social Media Marketing can span multiple departments. This is tricky because of the cross-functional collaboration aspect mentioned above - people will want to be clear on who 'owns' social, but at its absolute best version, social media touches multiple departments of a business. People from all those departments are contributing, sharing, and proposing new ways to connect with fans/followers.
Consider 'the online business card' concept. Even if you don't explicitly generate revenue from social media (many businesses don't), it's still your online business card. When a prospect hears about you, they'll go look at your website and channels. This is commonplace now. You need to have something there which conveys your service, product, and value. If you don't, potential prospects will move on to the next potential business partner. This is often a flaw in senior leadership thinking about social media: it might not directly make money, but it can indirectly lose you a lot of money.
What type of person should manage social media for you?
This could be its own blog post because it touches on various aspects of a recruiting, interviewing, and hiring process. For this post, though, we'll keep it relatively simple.
The conventional logic that many use is that you should hire a Millennial (someone roughly 20-32 right now) to do social media, because many of this generation grew up with digital/mobile/social and might be more effective at it.
That's not a bad idea, although obviously it varies person-to-person.
The one potential concern with Millennials, though, is that oftentimes they grew up using social on a personal level. A professional level, in terms of tone and alignment with business goals, can be very different. You need to watch out for that.
And of course, oftentimes people of the Gen-X/Boomer generation (34-60s right now) are the most willing to learn and grow for a company.
Who should call the shots on social media?
The most obvious answer here is "the head of marketing" or "the head of the department you're putting social in." From a hierarchy standpoint, that makes a lot of sense.
Consider this graph, though;
The majority of CMO's polled here either don't have a plan to implement an omni-channel strategy (1 in 4), are in early implementation (3 in 10), or 'plan to launch in a year' (3 in 10). In short: most CMO's don't fully embrace digital - which is logical, because it's very different than the traditional business models they came up in their industry with.
When you look at statistics like these, it makes sense to insert a middle-level manager in marketing who understands how social media works and have them call the shots in terms of social strategy and direction. Assigning that responsibility to the CMO means it potentially gets lost in their inbox, or in their slew of other responsibilities.
Everything will vary by your team, your industry, and your hiring needs/abilities - but I would recommend:
Make sure Social Media Marketing resides in the same department as content production, although it can fan out to other departments
Don't use a completely standard hierarchy approach to who becomes the primary decision-maker, because that can downshift social as a priority (and you can miss a business opportunity as a result)
There are plenty of opportunities to have the right people trained in social media.