When should we create a new social network website to sell a service or product?

Are you in the social network business? The answer to your question is that simple.  Social networks are unto themselves, businesses.  If you're in the business selling a product or service, why would you build a network?  Perhaps you're thinking that a social network is a way to acquire customers??  No.  A community is, perhaps, but not a network that you'd have to design, develop, operate, manage, and promote in addition to your product or service just to get it to work. Social networks are "marketplace" ventures in which the very value, the product they deliver, is the network of other people.  It's a bit of a catch-22... to get anyone to use your network, you have to get people / to get people, you have to have people using your network.  Isn't that a community?  Not at all, your existing customers or clients are your community, you don't need to build and operate anything distinct in that regard. So perhaps you're asking how to grow your community into something more?  To turn them into a resource that helps you sell your product or service?  You think, perhaps, that the people that gravitate around your product or service want to have friends, profiles, and contribute content? Do you know that?  Have you asked any of them?  Perhaps there is something simpler that they want - your engagement on Facebook, sharing pictures of your dog on Pinterest, or tweets about the weather.  I don't know, you don't know, that's the point; you said that of course you can consider an existing social network such as LinkedIn or Facebook - that very point affirms that you haven't asked your existing community what they'd actually want and whether or not a branded social network owned an operated by you is even something they'd use.  By the way, Facebook and LinkedIn are two VERY different networks and experience; at the very least, you should start by knowing where you should be therein. Sorry if that sounds terse, not my intent.  I'm trying to be blunt to help you focus.  Your question suggests you haven't even asked the people you'd hope use such a thing.  That's where to start, not here.  Then, even if they say they'd be interested in a distinct social network, realize that building one requires you enter a completely new business - are you prepared for those costs, human resources, development, etc. to do that?

Never

The social website effect on business is overestimated. I would not spent effort to create one more social network. It is a distraction. If you want to convince yourself, there is one not risky approach. Try some existing social site. Then count the clients this site gave you. Also count the time you spent managing your presence on that site. You will be surprised how ineffective it was.

There are a lot of factors to consider when thinking about starting a social media site for a specific product or service. Ask yourself: Who is my audience/target customer? Are they active on social media? How do competitors use social media to engage/convert customers in this space? Do I have the time and resources needed to create and support and online community with rich content and ongoing interaction? Can I clearly articulate/design the customer flow from social to other channels and track those metrics/results? Can I provide value to my customers or potential customers on a regular basis? Do you have a clear idea of what metrics will allow you to show success? (Beyond likes, follows, etc. and into actual revenue generation?) Ultimately, social can be a complementary layer on top of other digital tactics. The ability to use social to get potential customers into the top of your digital conversion funnel should not be underestimated when implemented intelligently and with clear purpose.