Is a person married to another if they both say "We accept each other as man and wife, having Allah as our witness," with no other witnesses present, and later having a celebration, telling everyone that they are married?
Praise be to Allah.
The Prophet (Peace & Blessings of Allaah be upon Him) said: "There is no marriage without a wali (guardian) and two witnesses." (It is a saheeh hadith because of corroborating reports: Irwa’ al-Ghaleel, no. 1858).
Imam al-Tirmidhi, may Allah have mercy on him, said: "The correct opinion is that narrated from Ibn ‘Abbas, who said that there is no marriage without clear evidence… Acting in accordance with that, the scholars among the Companions of the Prophet (Peace & Blessings of Allaah be upon Him) and the Taabi‘een and others who followed them said that there is no marriage without witnesses. (Jaami‘ al-Tirmidhi, 4/235).
If the couple referred to in the question did not follow the above, they must repeat the marriage, this time with a guardian and two witnesses.
And Allah knows best.