By offering shared computational facilities to which mobile devices can offload their computational tasks, the mobile edge computing framework is expanding the scope of applications that can be provided on resource-constrained devices. When multiple devices seek to use such a facility simultaneously, both the available computational resources and the available communication resources need to be appropriately allocated. In this manuscript, we seek insight into the impact of the choice of the multiple access scheme by developing solutions to the mobile energy minimization problem in the two-user case with plentiful shared computational resources. In that setting, the allocation of communication resources is constrained by the latency constraints of the applications, the computational capabilities and the transmission power constraints of the devices, and the achievable rate region of the chosen multiple access scheme. For both indivisible tasks and the limiting case of tasks that can be infinitesimally partitioned, we provide a closed-form and quasi-closed-form solution, respectively, for systems that can exploit the full capabilities of the multiple access channel, and for systems based on time-division multiple access (TDMA). For indivisible tasks, we also provide quasi-closed-form solutions for systems that employ sequential decoding without time sharing or independent decoding. Analyses of our results show that when the channel gains are equal and the transmission power budgets are larger than a threshold, TDMA (and the suboptimal multiple access schemes that we have considered) can achieve an optimal solution. However, when the channel gains of each user are significantly different and the latency constraints are tight, systems that take advantage of the full capabilities of the multiple access channel can substantially reduce the energy required to offload.