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 varepsilon. Such a guarantee is quite demanding, and recent works show that privatizing texts under local DP can only be done reasonably under very high varepsilon 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 varepsilon values, while still balancing privacy and utility. These findings highlight the importance of coherence in achieving balanced privatization outputs at reasonable varepsilon levels.