Well, there are many problems. Probably the biggest is they are a shadow market when in fact they should be traded on an open, regulated exchange. That would end many of the problems we currently have.
However, the second biggest problem we have no idea how to tax them. The problem is they could be conceivably be three things, although I think it boils down to two.
Are they insurance? In some CDS transactions, the person purchasing the contract is compensated for in cash. The person selling the contract pays the purchaser cash based upon the difference between a value specified in the contract and the then market value of the CDS. Essentially, the person purchasing the contract has shifted the risk (key legal term) to the seller, forcing the seller to pay x amount of dollars at a specific time. There is another key legal issue with risk distribution which raises some problems here, but I believe that proper regulation would prevent that from happening.
Are they an option? In other CDS transactions, the buyer will put a specific investment to the seller much like an option. In effect, the buyer pays for the right to say, "this is now worthless. You take it. By the way, you have to pay a previously agreed to price." This is essentially an option.
Further complicating the matter is the way CDS are documented. First, there is a "master agreement" between parties which essentially establishes an entire framework for how they will interact. Then each transaction can have its own terms and conditions. The point of this methodology is to promote the transaction process -- that is, to create an environment where people write a lot of transactions. These documents are available from the International Swaps and Derivatives Association.
I'm currently writing a paper on this -- which is why I bring it up. Unfortunately there are no clear answers -- just intellectual mud.