There are two different issues here that tend to get confused.
1) Any Chometz that was owned by a Jew over Pesach (your local supermarket that might be owned by a non-religious Jew). It is forbidden to derive any benefit from it even after Pesach. This is an issur m’derabonon. Regarding how to define ownership (the manager is Jewish, some of the shareholders are Jewish, etc.), speak to your local halachically competent rabbi.
2)Any chometz that was correctly sold by a Jew to a gentile over Pesach is %100 permissible to be eaten. Many people are of the custom not to sell actual chometz of their own unless a large monetary loss will be incurred through its disposal (a store owner or an expensive stash of whiskey). It is common, especially in Eretz Yisroel, to find those who will be stringent on themselves not to use chometz, even though it was properly sold, whose grain was ground before pesach (besides for flour that was especially prepared in a chometz free way – like that which is used for matzoh). You will find these people specifically looking for the label that says, “nitchan achar hapesach”. This is not an halachic requirement but rather a stringency mentioned by the Chazon Ish which many people have come to adopt.