As far as getting permission to re-marry is concerned, the Catholic Church holds all marriages to be valid until proven otherwise. A person married invalidly has to formally demonstrate that all prior marriages fall into that category before being permitted to attempt marriage again. (IOW, I don't mean that the Church can't say which marriages are invalid based on objective facts, only that the objective facts have to be provided for each case.)
There are marriages that are invalid with a flaw that is not correctable--say, if you murdered the former spouse in order to be able to marry them, or you are siblings, or one of you is still in a valid marriage to someone else--and then there are marriages that are invalid but have flaws that are correctable--for instance, if a Catholic married outside the Church without permission. Some defects don't require the full investigation by a tribunal that others do, which may be why there is the confusion about whether "an anulment is needed." If there is a defect of form--for instance in the case of marriages outside the Church and without permission--then former spouses do not need to be contacted to establish the defect, because the objective facts are in the public record.
If the flaw is correctable, then the couple can seek a convalidation. Convalidation is the exchange of vows between a couple whose marriage has not been recognized by the Church. For example, if Catholics marry "outside the Church," and the couple later wants to have their marriage "blessed by the Church," they come to the Church to exchange their vows and "convalidate the marriage."
This is different than when validly married Catholics reaffirm their vows to celebrate an anniversary, for instance, becuase the Church would not convalidate marriages that it already recognizes. If two Baptists were married in a Baptist church, the Church recognizes that as a valid marriage. If one or both of the couple joins the Catholic Church, their marriage would not be convalidated, as the marriage is already valid and sacramental.