Achieving differential privacy (DP) guarantees in fully decentralized machine learning is challenging due to the absence of a central aggregator and varying trust assumptions among nodes. We present a framework for DP analysis of decentralized gossip-based averaging algorithms with additive node-level noise, from arbitrary views of nodes in a graph. We present an analytical framework based on a linear systems formulation that accurately characterizes privacy leakage between nodes. Our main contribution is showing that the DP guarantees are those of a Gaussian mechanism, where the growth of the squared sensitivity is asymptotically $O(T)$, where $T$ is the number of training rounds, similarly as in the case of central aggregation. As an application of the sensitivity analysis, we show that the excess risk of decentralized private learning for strongly convex losses is asymptotically similar as in centralized private learning.