Do we have to eat meat that has been slaughtered by Muslims only?
What Allah forbids is slaughtering the animals pronouncing the name of any other than Allah, whatever the religious identity of the person is. It is equal whether the person who slaughters the animal is Muslim, or Christian, or Jewish; as long as that animal is slaughtered for the name of others than Allah, that animal is “ohilla lighayri Allah” and eating that meat is forbidden (haraam). Although the person is Christian or Jewish, that meat can be eaten as long as the animal is not slaughtered in the name of any other than Allah. That meat would be forbidden only if you are sure that it was slaughtered in the name of other than Allah.