Machine learning (ML)-based feedback channel coding has garnered significant research interest in the past few years. However, there has been limited research exploring ML approaches in the so-called "two-way" setting where two users jointly encode messages and feedback for each other over a shared channel. In this work, we present a general architecture for ML-based two-way feedback coding, and show how several popular one-way schemes can be converted to the two-way setting through our framework. We compare such schemes against their one-way counterparts, revealing error-rate benefits of ML-based two-way coding in certain signal-to-noise ratio (SNR) regimes. We then analyze the tradeoffs between error performance and computational overhead for three state-of-the-art neural network coding models instantiated in the two-way paradigm.