If they tell you the food is kosher, the halacha is they have ne'emanus and you can believe them. You don't need to start questioning which shitos they follow. If you become aware that something they serve is not up to your standard, you believe them and can no longer eat that item.
With due respect, this makes no sense to me, and it would behoove you to provide a source. I grew up on shlichus, and I can all but guarantee that you would not eat in someone's house in the area I grew up in, even if the person assured you the home was kosher. You would know that despite their best intentions, they simply aren't aware of the extensive halachos involved in keeping kashrus even close to your standards. My father had a strict "I don't eat in anyone's house" policy, which he still keeps to even though he lives in a frum neighborhood now, and it insults many people. But often the people most offended are the very people the policy is in place for. They believe they are keeping kashrus to the highest standards, but are simply ignorant about many aspects of kashrus. When someone shows you they don't even know that there is a question about tuna, and this is a major problem for you, I fail to see how neemanus solves the problem of them keeping standards that are well below par for you.