This is a very good question and one that musicians everywhere will have a different answer for. I start to answer this for myself by asking myself the following questions:
1. Can I play the left hand and the right hands separately from memory at both a slow and fast tempo (speed)?
2. Can I play the piece hands together from memory at both a slow and fast tempo?
3. Can I play the piece from multiple starting points from memory?
4. Have I executed accurately my plan for articulations, dynamics, and phrasing in the piece?
5. Do I make the melodies sing?
6. Have I put myself into the music?
The last question is perhaps the most difficult to answer. For me, it often comes about because the piece in question conjures up a setting, situation, story and/or a feeling and it is these things that I try to communicate through my playing. Sometimes I play with my eyes shut because I feel like the distance between myself and the music is decreased. In other words, I can feel more like I am part of the music when I shut my eyes and I am playing. Of course, when you shut your eyes, you are giving up the benefits of visual memory which can be extraordinarily important when executing large leaps but at times I do this anyways, even when I am performing, because I can find myself in the music more easily.