Building on recent development by Padakandla and Pradhan, and by Lim, Feng, Pastore, Nazer, and Gastpar, this paper studies the potential of structured nested coset coding as a complete replacement for random coding in network information theory. The roles of two techniques used in nested coset coding to generate nonuniform codewords, namely, shaping and channel transformation, are clarified and illustrated via the simple example of the two-sender multiple access channel. While individually deficient, the optimal combination of shaping and channel transformation is shown to achieve the same performance as traditional random codes for the general two-sender multiple access channel. The achievability proof of the capacity region is extended to the multiple access channels with more than two senders, and with one or more receivers. A quantization argument consistent with the construction of nested coset codes is presented to prove achievability for their Gaussian counterparts. These results open up new possibilities of utilizing nested coset codes with the same generator matrix for a broader class of applications.