Consider the similarity and the difference between the following English expressions:
- I and you met (or You and I met)
- I met you
- You met me
The first one handles the two people as a group and as the subject. The second one is describing from the viewpoint of I; you is the "destination/target" of meeting. The third one is the same construction but from the viewpoint of you.
Your third example does not sound quite right unless you add
私と, and if you do so, it will correspond to the first English example.
- (私と)あなたが出会う 'I and you meet'
For the second and the third of your examples, the subject is omitted, which is okay, and they both correspond to the second English example with a slightly different nuance.
と means 'with', whereas
に means 'to'. So the word-to-word translation will be something like:
- (私が)あなたと出会う '(I) met with you'
- (私が)あなたに出会う '(I) met to you'
They both may not sound grammatical in English, but that is how Japanese works. Whatever difference you can feel between
to is the difference between the Japanese sentences. In reality, the difference is so subtle so that you do not have to care. In practice, you can think that your second and third examples are interchangable.