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We present neutral atomic hydrogen (HI) observations using the Robert C. Byrd Green Bank Telescope (GBT) along the lines of sight to 49 dwarf satellite galaxy candidates around eight Local Volume systems (M104, M51, NGC1023, NGC1156, NGC2903, NGC4258, NGC4565, NGC4631). We detect the HI reservoirs of two candidates (dw0934+2204 and dw1238$-$1122) and confirm them as background sources relative to their nearest foreground host systems. The remaining 47 satellite candidates are not detected in HI, and we place stringent $5σ$ upper limits on their HI mass. We note that some (15/47) of our non-detections stem from satellites being occluded by their putative host's HI emission. In addition to these new observations, we compile literature estimates on the HI mass for an additional 17 satellites. We compare the HI properties of these satellites to those within the Local Group, finding broad agreement between them. Crucially, these observations probe a ``transition'' region between $-10\gtrsim M_V \gtrsim -14$ where we see a mixture of gas-rich and gas-poor satellites and where quenching processes shift from longer timescales (i.e. via starvation) to shorter ones (i.e. via stripping). While there are many gas-poor satellites within this region, some are gas rich and this suggests that the transition towards predominantly gas-rich satellites occurs at $L_{V}\sim10^{7}L_{\odot}$, in line with simulations. The observations presented here are a key step toward characterizing the properties of dwarf satellite galaxies around Local Volume systems and future wide-field radio surveys with higher angular resolution (e.g.~WALLABY) will vastly improve upon the study of such systems.