It depends on what you're looking to monetize it with.
One recommendation is affiliate marketing, because it happens under the table, in a sense.
For example, get the plugin provided at the bottom of this answer.
Browse amazon for products particular to your audience. If you're running a Facebook page and Twitter account for music-lovers, go to Amazon mp3s and get affiliate links for popular albums.
Post an engaging status/tweet ("You have to check these guys out:") followed by the affiliate link. You get money for each purchase.
Another option is fiverr.com (and other websites like it - microcommerce sites) where you can sell services for $5. It is common for people to buy tweets/posts from people with large social network accounts. For example: "I will Tweet your message to my 50,000 music-loving fans for $5".
There are many other ways to monetize, including partnerships with sites trying to promote content that your fans will enjoy, but these are two of the simplest. They don't require anyone else for you to get started.
Hope that helps! I'd love to talk more.
The answer to that question really hinges on what the niche of the social network is.
Let's say the niche is a about insects. Perhaps an alternative to advertising, would be to send out free bug spray to all members with a dominant bug spray company sponsoring the giveaway of their product.
Another example of a way to monetize a network without advertising is to ask for donations once every few months via an email broadcast and banners on your site for a limited time to promote the donation.
There are a million ways to monetize any website, it all comes down to identifying your audience and what state of mind they are in while navigating your webpage.