I think δικές and μας must always be next to each other. I think the attribute comes before the noun it describes.
To have δικές μας after πατάτες, the attribute function has to vanish; it could be done by inserting a verb, as in οι πατάτες είναι δικές μας.
Disclaimer: I am not a native speaker of Greek, but I have been living in Greece for several months and only noticed the attribute in front of the noun that is described.
Use δικοί with masculine (plural) nouns. If you're talking about our own roads, this would be οι δικοί μας δρόμοι. (The same οι is used for both feminine and masculine, plural.)
In Greek, the article οι, the word δικές and the noun πατάτες must agree — they are all feminine plural. It may be a little confusing at the beginning, because the ending is different (οι, ες), but they agree grammatically.
To contrast with the example about roads (οι δικοί μας δρόμοι) I'll give another one, which is also for masculine plural: οι δικοί μας άντρες (our own men/husbands) which is also grammatically correct, but δικοί and άντρες look different because of the different endings. Still, both of these words are in the masculine plural form.