Great question. A couple of additional thoughts from my end. They're all over the place but hopefully they help. In general I agree with others that you should hire a "growth hacker" and expect them to be a game changer over night. In fact I tell most people on the growth side not to take a job with a company advertising a role as a growth hacker because that company clearly has no idea what they need to actually grow.
But that said, someone must own growth. This can be a marketing person, a founder, the CTO, a product manager, but just like someone in finance owns how much money is in the bank, someone on the team should own measuring and managing the growth of the company.
Startups are generally in two modes when it comes to growth: channel/lever discovery and channel/lever optimization. In the discovery phase you probably want more of a generalist who can work across different vectors of the business to run experiments and find signal for potential growth opportunities. When signal arises, then you want to double down on those opportunities and ultimately, as you scale specialize growth roles around people with expertise in various systems/distribution channels.
The exact person to fill the growth role is really going to depend on the strengths of your existing team. So if you lack growth expertise in your product team, you might decide to bring someone on with that expertise. Most growth roles sit within the product organization at larger companies. So you might hire a product manager who leads growth initiatives if your product team is not particularly strong at growth. Or if you feel you've got good signal on product/market fit and need to figure out which channels are going to work, you might bring in a digital marketer who can lead that discovery and optimization process.
Just a note on the same thing, different day argument. First, if people want to spend time parsing what is a growth hacker, or what is or isn't growth hacking, they are welcome to spend their time on that nuance. To me, if everyone spent more time focused on growth and less on the hacking part, we'd be better off. I just don't get why people get angry about it -- the fact that the term and the renewed focus on growth has spurred better awareness of its role in the success of a company is a good thing.
Growth hacking is certainly a buzzword but the focus on experiment driven growth is different than what many people in the traditional marketing role practice. Talk to a marketer about what a growth person does and they will often cite user acquisition strategies through traditional channels like paid search, email, Facebook ads, etc. Not that these aren't important, but if you look at the fastest growing companies -- the $1B plus club -- very few got initial traction with these traditional channels. (Of course you could make the argument that great marketers do focus on growth and there's just a lot of bad marketers, and I wouldn't disagree with you.)
Lastly in many larger companies with growth teams, they are independent from marketing. Most growth teams consist of engineers, product people, data scientists, etc. who are focused on product usage that results in growth. It looks much different than a marketing team who traditionally couldn't have access to the product and was relegated more to top of funnel activities.
As you can see I enjoy talking about this subject so feel free to follow up with any additional questions.