Is paying a sum of money to someone who sought to employ me considered bribery, and is it due to the first seeker or the second after the second one completed the employment?

If the second employee facilitates your employment in a permissible job without deception, fraud, violating conditions, or infringing on anyone's right, then the promised amount for him is permissible and is considered a 'ju'ala' (a stipulated reward for a service), not a forbidden bribe. As for the first employee, he is not entitled to anything because he did not complete the employment. However, if the employment involves deception, fraud, violating work conditions, or wasting a right, then mediation and payment of money are not permissible, and that is considered a forbidden bribe.