The efficiency of Public Key Encryption (PKE) and Key Encapsulation Mechanism (KEM), and in particular their large ciphertext size, is a bottleneck in real-world systems. This worsens in post-quantum secure schemes (e.g., lattice-based ones), whose ciphertexts are an order of magnitude larger than prior ones. The work of Kurosawa (PKC ’02) introduced multi-message multi-recipient PKE (mmPKE) to reduce the amortized ciphertext size when sending messages to more than one recipient. This notion naturally extends to multi-message multi-recipient KEM (mmKEM). In this work, we first show concrete attacks on existing lattice-based mmPKE schemes: Using malicious public keys, these attacks fully break semantic security and key privacy, and are inherently undetectable. We then introduce the first lattice-based mmKEM scheme (thereby mmPKE) that maintains full privacy even in the presence of maliciously-generated public keys. Concretely, the ciphertext size of our mmKEM for 100 recipients is \({>}10\times \) smaller than naively using Crystals-Kyber. We additionally show a similar efficiency gain when applied to batched random oblivious transfer, and to group oblivious message retrieval. Our scheme is proven secure under a new Module-LWE variant assumption, Oracle Module-LWE We reduce standard MLWE to this new assumption for some parameter regimes, which also gives intuition on why this assumption holds for the parameter we are interested in (along with additional cryptanalysis). Furthermore, we show an asymptotically efficient compiler that removes the assumption made in prior works that recipients know their position in the list of intended recipients for every ciphertext.