If the purchases are real purchases then you definitively wanna do server side management of what a player owns. In offline mode you just don't get the skins. Otherwise there will be ways players come up with to get stuff for free.
The key question here is that do you want to make the game "always online" or not.
If it's not then you're bound to store some data locally that may be manipulated by the player.
There are ways like check-sums and encryption that can make this difficult enough for most players not to bother though in which chase I'd be more worried about memory editing which is a problem with server side management as well if the game is still ran locally.
(modified high-scores etc. being sent to server)