Antibodies are widely used as therapeutics, but their development requires costly affinity maturation, involving iterative mutations to enhance binding affinity. This paper explores a sequence-only scenario for affinity maturation, using solely antibody and antigen sequences. Recently AlphaFlow wraps AlphaFold within flow matching to generate diverse protein structures, enabling a sequence-conditioned generative model of structure. Building on this, we propose an alternating optimization framework that (1) fixes the sequence to guide structure generation toward high binding affinity using a structure-based predictor, then (2) applies inverse folding to create sequence mutations, refined by a sequence-based predictor. A key challenge is the lack of labeled data for training both predictors. To address this, we develop a co-teaching module that incorporates valuable information from noisy biophysical energies into predictor refinement. The sequence-based predictor selects consensus samples to teach the structure-based predictor, and vice versa. Our method, AffinityFlow, achieves state-of-the-art performance in proof-of-concept affinity maturation experiments.