Physics-informed neural networks (PINNs) and neural operators (NOs) have emerged as powerful machine learning frameworks for solving partial differential equations (PDEs), where physical laws are embedded directly into the model architecture or training objective, enabling mesh-free solution representations. Despite their flexibility, accurately handling interface problems remains challenging, primarily due to the difficulty of enforcing discontinuous interface conditions. Standard soft-constraint approaches incorporate interface conditions through penalty terms in the loss function, which often compete with PDE residual minimization and lead to degraded accuracy and stability near interfaces. While several hard-constraint formulations have been proposed to address this issue, most are limited to continuous systems and do not directly accommodate discontinuous interface conditions. In this work, we introduce two hard-constrained strategies for solving elliptic interface problems that are applicable to both PINNs and neural operators. The first strategy, referred to as the windowing approach, embeds boundary and interface conditions directly into the solution ansatz, ensuring exact satisfaction of continuity and flux balance conditions by construction. The second strategy, termed the buffer approach, introduces auxiliary buffer functions that locally compensate neural network outputs at sampled points to enforce boundary and interface constraints. Both approaches effectively decouple the enforcement of interface conditions from PDE residual minimization, thereby eliminating competition between multiple loss terms. We evaluate the proposed methods on representative one- and two-dimensional elliptic interface problems. Numerical results demonstrate that both hard-constrained approaches achieve significantly higher accuracy and improved stability compared to conventional soft-constrained formulations, particularly in the vicinity of interfaces. We further compare the performance characteristics of the windowing and buffer approaches and discuss their extension to neural operators. These results highlight the potential of the proposed methods for challenging interface-dominated physics problems. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 and was supported by Laboratory Directed Research and Development funding under project 25-ERD-052. LLNL-ABS-2014782.