Choosing words is part of the process of realization, of defining our thoughts and feelings for ourselves, as well as for those who hear or read our words.
We had different ideas of the the of-phrase. I thought it was a postmodifier of the word 'process', while she said it should modify 'realization'. And since she is not a native speaker, I thought I'd better seek help here. Any reply would be appreciated. Thanks!
'of' is a preposition, and prepositions require an object. In , the 1st 'of' takes 'the process of realization' as its object, and the second 'of' takes 'realization' as its object.
 . . . part of the process of realization. . . .
Now, given those structural relationships, this is how your example sentence reads,
". . . part of the process of realization, [part of the process] of defining our thoughts.