The problem of reliable/secure all-to-all communication over low-degree networks has been essential for communication-local (CL) n-party MPC (i.e., MPC protocols where every party directly communicates only with a few, typically polylogarithmic in n, parties) and more recently for communication over ad hoc networks, which are used in blockchain protocols. However, a limited number of adaptively secure solutions exist, and they all make relatively strong assumptions on the ability of parties to act in some specific manner before the adversary can corrupt them. Two such assumptions were made in the work of Chandran et al. [ITCS ’15]—parties can (a) multisend messages to several receivers simultaneously; and (b) securely erase the message and the identities of the receivers, before the adversary gets a chance to corrupt the sender (even if a receiver is corrupted). A natural question to ask is: Are these assumptions necessary for adaptively secure CL MPC? In this paper, we characterize the feasibility landscape for all-to-all reliable message transmission (RMT) under these two assumptions, and use this characterization to obtain (asymptotically) tight feasibility results for CL MPC.