Not on every stone, and the makers' own words differ by stone type. Case designs its honing oil for use with sharpening stones and recommends it on its Bench-Top and Aluminum Oxide Oilstone pages. Buck recommends using natural stones wet, with water, water-based honing fluid or petroleum-based honing oil, and says diamond sharpeners can be used dry or wet but not with petroleum-based oil.
Case ties its honing oil to its stones. The oil's own page and Case's kit pages say it is designed for sharpening stones, and the Bench-Top Sharpening Kit and Aluminum Oxide Oilstone pages each recommend Case's honing oil when sharpening. Case's Hard Arkansas Pocket Stone page names its stone an oilstone but prints no oil instruction of its own.
Buck treats diamond separately. Buck's how-to page says "Diamond Sharpeners can be used dry or wet" and "Use water or water-based honing fluid, not petroleum-based oil, as a lubricant." That is Buck's instruction for Buck's diamonds; if you are unsure whether a lubricant counts as petroleum-based, ask Buck before it goes on one. For the steps on a diamond plate, read how to use a diamond sharpener.
For natural stones Buck names three lubricants, not one: water, water-based honing fluid or petroleum-based honing oil. In its fine-sharpening stage Buck adds that sharpening "will require some clean up, so be generous with the honing fluid."
The cited Case and Buck pages do not explain changing a stone from oil to water or back. Use the lubricant the stone's maker names and ask that maker before switching. How many stones a kit needs is answered in whether different grits are needed.