After being legally divorced on paper, should he notify the home office about his divorce before I file for him?

Although not entirely clear from your question, I presume that you are a U.S. citizen who wishes to sponsor your husband for immigration benefits in the U.S., and your husband currently is in Europe, where he became divorced from his first wife. If that is accurate, you and your husband may apply for consular processing for him to come to the U.S. and then you may petition for him to adjust status to become a Lawful Permanent Resident (to get a "Green Card"). Part of those steps will include providing documentary evidence that his previous marriage was lawfully terminated through divorce. It would be wise for you and your husband to work with an immigration attorney for this process. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.