This is actually in answer to a PM that I received. However, since other might have the same question, I am answering it here.
There are several options here depending upon whether or not you want to save some of the PM's. They all hinge on whether or not you want to save what you have received and/or sent (being a moderator, I save all of them.)
First, I would recommend cleaning out your Sent box. When you go into Private Messages, you notice that directly under the bar that indicates how much mail you have, there is a command that says "Jump to Folder". Click on that to go to either "Sent Items" or "Archive". Once you have selected the appropriate folder, then you hit the "Go" button to the right. You can then click the boxes to the right of each message you want to clean out of your mail box. After clicking on these boxes, go to the box at the bottom right that says "Selected Messages". Scroll among the choices offered to determine what you want to do. Choices include: move to another folder (usually used for moving to archive folder), delete, two mark functions, and three download functions (I personallly use the Download as CSV so that I can load them into a spreadsheet for later processing and searching if necessary). One comment - if you download the messages, you must still delete them to free up space. Downloading does not remove them. Once you have cleaned out your "Sent" box you can move back up to "Select folder" to do the same with either you inbox or archive. Personally, when I download, I tend to do it by weeks at a time and do not download anything in the current week if I can help it.
I hope this answered the question.