The Church presumes the marriages of two unbaptized people or a baptized person and an unbaptized person to be valid. If both partners are baptized, the Church presumes the marriage to be valid and sacramental. Only Catholics are formally required to observe the Catholic rite of marriage, so non-Christian marriages can be valid and non-Catholic Christian marriages can be valid and sacramental. Since the Church wants to take a look at anything that looks as if a marriage took place in order to determine the freedom of the parties to marry, any previous marriage should be put before a marriage tribunal before a subsequent marriage in the Catholic Church is attempted.
As a side note, even for Catholic marriages a priest is not required to witness the ceremony. If the couple is getting married in a nuptial liturgy outside of Mass, a deacon may preside at the wedding.