The only marriages that need to be annulled are those that are considered "official" by the church. Now, only one party needs to be Catholic to considered an official marriage. A priest needs to be in attendance at the ceremony, and counsel with the parties before the wedding.
Once the annulment is official, a Catholic is free to marry a Catholic or non-Catholic, so long as the wedding is conducted with the approval and attendance of a priest.
These are the rules as I understand them. I believe they are correct, but I could be wrong.
