Cell-free massive multiple-input multiple-output (mMIMO) is a promising technology to empower next-generation mobile communication networks. In this paper, to address the computational complexity associated with conventional fingerprint positioning, we consider a novel cooperative positioning architecture that involves certain relevant access points (APs) to establish positioning similarity coefficients. Then, we propose an innovative joint positioning and correction framework employing multi-agent reinforcement learning (MARL) to tackle the challenges of high-dimensional sophisticated signal processing, which mainly leverages on the received signal strength information for preliminary positioning, supplemented by the angle of arrival information to refine the initial position estimation. Moreover, to mitigate the bias effects originating from remote APs, we design a cooperative weighted K-nearest neighbor (Co-WKNN)-based estimation scheme to select APs with a high correlation to participate in user positioning. In the numerical results, we present comparisons of various user positioning schemes, which reveal that the proposed MARL-based positioning scheme with Co-WKNN can effectively improve positioning performance. It is important to note that the cooperative positioning architecture is a critical element in striking a balance between positioning performance and computational complexity.