Although thousands of long non-coding RNAs (lncRNAs) are encoded in mammalian genomes, their mechanisms of action are poorly understood, in part because they are often expressed at lower levels than their proposed targets. One such lncRNA is Xist, which mediates chromosome-wide gene silencing on one of the two X chromosomes (X) to achieve gene expression balance between males and females. How a limited number of Xist molecules can mediate robust silencing of a much larger number of target genes while maintaining specificity exclusively to genes on the X within each cell is not well understood. Here, we show that Xist drives non-stoichiometric recruitment of the essential silencing protein SHARP (also known as SPEN) to amplify its abundance across the inactive X, including at regions not directly occupied by Xist. This amplification is achieved through concentration-dependent homotypic assemblies of SHARP on the X and is required for chromosome-wide silencing. Expression of Xist at higher levels leads to increased localization at autosomal regions, demonstrating that low levels of Xist are critical for ensuring its specificity to the X. We show that Xist (through SHARP) acts to suppress production of its own RNA which may act to constrain overall RNA levels and restrict its ability to spread beyond the X. Together, our results demonstrate a spatial amplification mechanism that allows Xist to achieve two essential but countervailing regulatory objectives: chromosome-wide gene silencing and specificity to the X. This suggests a more general mechanism by which other low-abundance lncRNAs could balance specificity to, and robust control of, their regulatory targets.