I understand. I don't think it would make you look bad to tell your friend that the person he recommended flaked out. If that sours the relationship, then what kind of friend is he? That doesn't sound like someone I'd want to do business with. You don't need to rub it in his face, but mention it to him so you don't have 800lb. gorillas in the room (who knows what the designer is telling him about you, right?). Make it known that you are out $X due to the irresponsible behavior of a designer he recommended. You're not upset at him, you just want to let him know. Plain and simple.
In the meantime, dispute the charge on Paypal and, if you paid with a credit/debit card, do a chargeback with your card company, too. It's not a big deal, both Paypal and your card company process thousands of these every day. Paypal is more likely to side with you if they've already lost the money from your card company, so I'd start with them. I've dealt with the exact same situation as you and this is what I did. It worked. Even if you paid with Paypal funds (not on a card), dispute the charge with Paypal.
Take the time to write a clear description of the reason for your dispute on the PayPal "resolution center," complete with quotes of what the designer said. So if you have in writing that he agreed to a deadline, say "John Doe, the designer, promised 'I will delivery the work product by August 1st' in an email to me dated July 10th. Today is August 10th and I haven't received the work or any communication from him." When I wrote my dispute, I copied and pasted this same description of events to my card company for their chargeback form. (Kill two birds with one stone.)
Finally, may I suggest something to you? I've hired many freelancers. I don't pay people based on the project any more. I also don't pay people upfront any more. The freelancer does the work for today, sends me the work and their hours, and gets paid. It keeps them on a really short leash, and I'm never at risk of losing a penny. If I don't like the work someone is doing, or if I can see they move slowly, I can pull the plug right away. So instead of saying, "Design all 20 screens of this web app for me" just say, "Just design the login screen of this web app for me today."