A boy and a girls wirte 3 times on a paper with their signatures that they both accept each other with out any other witness and if the same persons introduce each other as a wife husband to others.. do such thing can count as "Nikah"
Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
It is a condition for the validity of Nikah (marriage) that the witnesses (two Muslim male witnesses or two female and one male witnesses) be present in the same place the marriage offer and acceptance take place.
Allah Ta'ala says,
"And get two witnesses out of your own men. And if there are not two men (available), then a man and two women, such as you agree for witnesses, so that if one of them (two women) errs, the other can remind her." (2: 282)
The wisdom for having witnesses for Nikah (marriage) and to announce it:
All the prophets (AS) and religious leaders are agreed that the marriage must be announced so that everyone may distinguish between it and adultery. So, the witnesses are appointed and it is reasonable that it should be further publicized through Walimah. People should be invited to it and it should be made known so that other people may learn of it and that there should be no misgiving later on. (The Wisdom Behind The Commands Of Islam, By Moulana Ashraf Ali Thanwi (RH)- Pg, 228)