The default behavior for PM's is that they do NOT automatically save a copy in your Sent folder. There is a check box you have to check if you want your outbound message saved to your Sent folder.
Please note that I don't send too many Private Messages, but the above is how it worked last time I sent one. :)
I have occasionally had stretches of time (a few hours), from time to time, where I couldn't send Private Messages. I just figured it was due to some sort of either maintenance at their end, or some issue that they eventually picked up on and reset, but each time - some time later - I was again able to send private messages.
Bob Bone