How do player to player purchases work?
say for instance and auction?

When you enter torishop http://forum.toribash.com/tori.php on the left are some options such as transfer toricredits, transfer items and safe trade. You can just talk out the deal with another player and either use safetrade or just send the items and credits each side should get.
Aside from that there's torimarket, you can click on the tab on top of forums and look for items sold by other players. You can put your own items on the market by going into your deactive inventory and clicking "edit" or on the icon of the item you wanna sell and giving it a price. It will be placed into market inventory and as soon as somebody decides to buy it, you get your money.