Cultural networking is an interesting idea! I think that a lof of that happens on regular social networking sites like Facebook. People often speak in different languages or discuss cultural phenomenon, and others get curious and check it out.
Cultural networking on a cultural level occurs most visibly on community-created content Web sites. At such sites, networking is available through comments sections; through private message capabilities; through slideshow and video uploading features; and through individual content written for a public audience.
The most successful cultural networking, community-created content site to date is Associated Content. Begun by a couple of guys at home (isn't that always the way) in Denver, CO, Associated Content (affectionately known as AC) has grown to phenomenal proportions; has contributors from all over the world; has experts in their fields (like climate physicists and financiers); and has ordinary individuals who have something to say. AC was recently purchased by Yahoo!, so those guys who began at home are now millionaires--and probably hosting a new project.
Two other similar sites are Hub Pages and Examiner. Hub Pages is like AC in that no special application, category, or acceptance is required to join and contribute, Examiner.com is different in that one applies for acceptance and is restricted to one's own city and a specific topic. To clarify, whereas one can write just about whatever one wants on AC and Hub Pages (within some limits), one writes only what one is qualified and accepted for and what reflects one's own locale on Examiner. There are fringe benefits to these sites. As a writer's content receives readers, what is called pay-per-view income is generated and may result in residual income.
There are some (a few) exciting success stories from AC, such as the Florida Philosophy and English teacher who was downsized and, being unable to quickly find another job, began writing about twenty articles a day for AC. Within a year he had a more than decent residual income, had been invited to speak on radio about some of his articles, got commissioned jobs to write specialized content, and established himself as a successful free lance writer. Another success story is of an expatriate American living in Europe who began writing witty travel articles on AC about the place she lived and the places she visited. In about a year's time, she had a job writing a travel column for a magazine or two and at the end of about two years was approached to write a travel book.
A cultural social networking site is very unique and thus not very common, you can checkout PiniWiki - http://www.piniwiki.com.
Sorry, I don't know of any other site, but you can try a google search for cultural networking