This paper investigates the network load balancing problem in data centers (DCs) where multiple load balancers (LBs) are deployed, using the multi-agent reinforcement learning (MARL) framework. The challenges of this problem consist of the heterogeneous processing architecture and dynamic environments, as well as limited and partial observability of each LB agent in distributed networking systems, which can largely degrade the performance of in-production load balancing algorithms in real-world setups. Centralised training and distributed execution (CTDE) RL scheme has been proposed to improve MARL performance, yet it incurs -- especially in distributed networking systems, which prefer distributed and plug-and-play design schemes -- additional communication and management overhead among agents. We formulate the multi-agent load balancing problem as a Markov potential game, with a carefully and properly designed workload distribution fairness as the potential function. A fully distributed MARL algorithm is proposed to approximate the Nash equilibrium of the game. Experimental evaluations involve both an event-driven simulator and a real-world system, where the proposed MARL load balancing algorithm shows close-to-optimal performance in simulations and superior results over in-production LBs in the real-world system.