This is the premise behind Tribe.net, BuddyBridge, and similar social networking sites. That is, you go to the site and find out if friends of your existing friends live in your new city (or the city you plan to move to). You email your potential new friends, let them know who you know in common, and ask to meet in a public place. If you hit it off, voila! new friends!
Friend referral sites are not meant, necessarily, to be dating sites. But if it happens, dating is OK, too. Tribe.net, for example, specifically protects user privacy. If people have a "first degree" or direct relationship to you, they can see your first and last name. If they only have a "second degree" (presumably friend of a friend), then they can see your first name and the first initial of your last name. If they only have a "third degree" connection, they see your first name only. So you control how much a stranger knows about you. Also, on Tribe.net, you don't have to disclose either your gender or your age unless you choose to.
PalJunction is another social networking site which allows you both to make new friends and to possibly date those new friends. Or have a business relationship with the new friend or simply buy a used car from your new friend. It's up to you.
Some friend referral sites (like LinkedIn and Ryze) are primarily for making new business friends to do business with as customers and/or hire as employees, contractors, consultants, etc. In other words, business networking.
All of the sites listed below allow new people to sign up and start inviting their friends to sign up (at least as of today, when I checked their sites). If existing friends decide to accept your invitation and also sign up and then invite all of their other friends, your network could grow quickly. Other sites, on the other hand, like Everyone's Connected require an existing member to "vouch" for you to complete your registration. (Worldshine/Six Degrees requires an invitation from an existing member, meaning no self-referrals at all.)
Meeting new friends via mutual friends has been something people have been doing forever. Now you can possibly do it faster on the Internet!
For more information see: