That would fill your email inbox up in no time.

If you have a busy site, potentially hundreds of PMs are sent a day - you want all of them to be copied and sent as an email to you?
Also, this could be a breach of your member's privacy as well - after all, PMs are called "private messages" for a reason.
Remember, if a member is doing something illegal using PMs, you have no control over it, and you should not be liable, unless you find out about it, and then you have a duty to stop this person as best you can.
There is a hack that allows you to read the PMs of individual members - I personally do not feel it is ethically correct, but you can install that if you wish.