A quantum network promises unconditionally secure transmission of data between its nodes by utilizing its ability to distribute entanglement across distant nodes. However, for most practical purposes, the distribution of entanglement is affected by environmental noise. We study the limits of secure communication between two parties sharing an arbitrary bipartite state or an arbitrary quantum channel under the one-way local operations and classical communication (one-way LOCC) setting, particularly for the non-asymptotic case. We use the ideas of unextendibility of entanglement to quantify the resourcefulness of a bipartite state or a quantum channel for forward-assisted private communication between its bearers, which we use to establish limits on the number of secret bits that can be established between the two parties either exactly, or probabilistically, or approximately. Our results surpass several previously known limits on secure communication under the considered setting. Additionally, several bounds presented in our work are efficiently computable, including the bounds on the one-shot private capacity of a channel, which are the first efficiently computable bounds on these quantities to the best of our knowledge.