Very sorry to hear this. The exact same thing happened to a friend of mine and she ended up paying for her ex' loan in full.
Surely not what you want to hear right now, but when you sign as a guarantor for a loan, you really should expect to get stuck with it in the end. Why would her ex even need a guarantor for a loan of 10k?
The bank can basically choose from whom to collect the money, so they will collect it from your girlfriend, but she could try to get it back from him. If he needs a guarantor to borrow 10k and is currently in a non-EU country, my hopes in that direction would not be too great...