Are there "right" finger positions to use in a piece?

When I first started with a piano teacher years ago, she always told me what finger should be on what key. But since I've stopped taking lessons about 3 years ago, I've found I just use the finger that feels most natural. Are pieces designed so that they automatically incorporate this into how they are played or does everyone do things subtly different. I mean, how am I supposed to know if what feels comfortable to me is the same for someone else (that is, unless I watch them play every piece I do)? Is one person's technique better than the others or does this even happen?