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Many works at the intersection of Differential Privacy (DP) in Natural Language Processing aim to protect privacy by transforming texts under DP guarantees. This can be performed in a variety of ways, from word perturbations to full document rewriting, and most often under *local* DP. Here, an input text must be made indistinguishable from any other potential text, within some bound governed by the privacy parameter π. Such a guarantee is quite demanding, and recent works show that privatizing texts under local DP can only be done reasonably under very high π values. Addressing this challenge, we introduce **DP-ST**, which leverages semantic triples for neighborhood-aware private document generation under local DP guarantees. Through the evaluation of our method, we demonstrate the effectiveness of the *divide-and-conquer* paradigm, particularly when limiting the DP notion (and privacy guarantees) to that of a *privatization neighborhood*. When combined with LLM post-processing, our method allows for coherent text generation even at lower π values, while still balancing privacy and utility. These findings highlight the importance of coherence in achieving balanced privatization outputs at reasonable π levels.