Does the dialogue that took place between the questioner and her colleague, which included a marriage proposal and her acceptance, constitute a valid marriage that requires her to be considered his wife, and are the men who overheard the dialogue considered witnesses to this marriage?

It is not permissible for a woman to work in a place where she mixes with men, except out of a recognized necessity. Nor is it permissible for her to joke with a non-mahram man or to form a relationship with him. Marriage has known pillars: the presence of the woman's guardian along with two just witnesses, a dowry, and a form of expression indicating the contract. What transpired does not constitute a valid marriage due to the absence of a guardian, a proper form of expression, or witnessing. A previously married woman (thayyib) does not marry herself; rather, she authorizes her guardian to marry her. A marriage contract is not valid without the presence of the woman's guardian. Therefore, the marriage was not concluded by the conversation that took place, and it is obligatory to cut ties and repent.