I have discovered a hacky way to send Chess.com private messages from the iPhone/iTouch. It's not perfect, but better than nothing.
The basic idea is that when you go to the "Send Message" screen, while the page is loading, it intially just shows a normal textarea instead of the "rich-text" thing that is loaded over the top of it. All you have to do is hit the "stop" icon (the small "x" on the address bar) in mobile Safari before the page finishes loading. Then you can use the regular old textarea to enter your message.
Unfortunately due to page styling the text will all be invisible, so you can't really expect to type out a long letter. If you just need to send a sentence or two, however, it should be manageable. Also, if you are replying to a message, don't forget that the text from that message will be pre-loaded into the textarea. I'd suggest just doing a "select-all" and deleting it to make things easier, or you can try to get your cursor to the beginning to insert your message at the top as per usual.
Finally, when you hit "Send Message" the page will first be reloaded (let it load all the way). You should see the message you typed in the rich-text area, now visible, so you can check it for any grievous errors before sending it off. Hitting the "Send Message" button again will send your message on its way.
Hope this helps some people out!