There are two variants of the chicken-and-egg problem:
What are general strategies for solving either (or both) variants of the "chicken-and-the-egg" scenario?
To spur discussion, I'll note that I've seen conflicting answers on this site:
This is a general, strategic question. It can sometimes be useful to think top-down from these types of questions to generate opportunities, in comparison to the bottom-up questions to solve specific problems.
Here are a couple ideas. I'm not sure about the specifics of what you're trying to do, so they may or may not be apt :)
One idea may be to simulate one side of the transaction. A friend of mine sells fashion merchandise to large department stores. However, to get into these department stores the stores require merchants to participate in trunk shows first. A trunk show is a temporary display setup in stores to test to see how your product does in various locations. What my friend did was essentially paid people to go into the store to buy the product.
The 99designs idea is pretty much the same thing, just pay people to do one side of the transaction until it becomes self sustaining.
Just to add a few ideas that might apply in some cases:
By right there is no real definitive answer to the Chicken and Egg problem else the problem wouldn't exist would it? I hate the C & E issue because I have seen people argue and discuss the problem in circles without ever making a choice.
And that is my answer to the problem. Make a choice! To escape the endless cycle of blaming chickens and eggs, just pick one to make a meal out of. Come up with several viable options as Greg has and make a call.
In the case of attracting buyers and sellers, I would say target sellers first. You won't have an audience, but many sellers (especially small time ones) are always looking for new (mostly free) outlets to sell their goods and services. So you may have to offer free service to the first group of sellers until you start attracting a crowd. After you have a pool of sellers, start marketing to the buyers.