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Large corporations, government entities and institutions such as hospitals and census bureaus routinely collect our personal and sensitive information for providing services. A key technological challenge is designing algorithms for these services that provide useful results, while simultaneously maintaining the privacy of the individuals whose data are being shared. Differential privacy (DP) is a cryptographically motivated and mathematically rigorous approach for addressing this challenge. Under DP, a randomized algorithm provides privacy guarantees by approximating the desired functionality, leading to a privacy–utility trade-off. Strong (pure DP) privacy guarantees are often costly in terms of utility. Motivated by the need for a more efficient mechanism with better privacy–utility trade-off, we propose Gaussian FM, an improvement to the functional mechanism (FM) that offers higher utility at the expense of a weakened (approximate) DP guarantee. We analytically show that the proposed Gaussian FM algorithm can offer orders of magnitude smaller noise compared to the existing FM algorithms. We further extend our Gaussian FM algorithm to decentralized-data settings by incorporating the CAPE protocol and propose capeFM. Our method can offer the same level of utility as its centralized counterparts for a range of parameter choices. We empirically show that our proposed algorithms outperform existing state-of-the-art approaches on synthetic and real datasets.