We insured our dog in Sweden, where it's very common, and it paid for an operation last winter that would have been a real shock otherwise. The key is to insure while the dog is young and healthy, because anything that happens before you take the policy is usually excluded later.
We never had pet insurance in Poland, because vet costs there were manageable. Here in Ireland the first vet bill for a minor ear infection made me reconsider everything. Our dog is five and healthy, but I'm now wondering whether we should insure him before something big happens.
Do you have pet insurance in your new country? Did it pay out when you needed it, and was it worth it compared to putting money aside?