That's the kind of nonsense that LLC vendors spout.  The fact is, if the LLC is sued, the owner will be sued.  It's difficult to imagine a situation where one would be liable and the other, not.  Once the owner has a judgment against him, all of his assets can be attached, including the other LLC.  

For small business, the best defense against liability is a good insurance policy.  The advantage is often the payment of legal fees.  Otherwise, even if the business wins, it incurs losses that can shut it down.  Not that I know of a good source for insurance against ASCAP and BMI claims.