OK, I'm not a professional, just an interested amateur in the history of technology, so if this answer is insufficiently thorough I apologise.

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It varied with time, place and technology, as you would expect for something that had been invented in the 18th century, spread widely and is still in (limited, as far as lighting goes) use now.

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One thing that needs to be understood about 19th century gas is that almost no gas used in the 19th century was natural gas we're most familiar with now. It was a family of various synthetic products from quite simple coal or wood distillate (the original coal gas) to a product of very sophisticated "complete gasification" process similar to modern synthetic gas, to acetylene, to things that weren't really what we would call a fuel gas at all ("Air-gas" was essentially a mixture of air and fumes of some liquid fuel.)

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The common raw materials for lighting gas had been coal, wood (less desirable, as wood gas doesn't burn very brightly,) A specific grade of oil shale/bituminous coal named cannel coal** was used for production of high-quality lighting gas and finally, oil, though expensive, was used in several processes either on its own or as an additive to boost the final product quality.

**Late edit: It appears I was a bit mislead about nomenclature of oil shales and bituminous coals here... Cannel coal can be referred to as both shale and coal, and often is in sources, but it's only true of that specific kind of coal, not all bituminous coal is also a shale.

If you're interested in contemporaneous descriptions, 1911 Britannica has two very long articles with detailed drawings of various lamps and gasworks equipment:

https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Lighting

https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Gas (the last part of the article is about synthetic lighting and industrial gas specifically)

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The business structure also varied, from small competing privately owned gasworks each with their own pipelines serving a city, to what was very close to modern utilities. In fact several modern utilities most notably British Gas and PG&E started as synthetic gas works operators.

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Safety and environment-wise, although they didn't store vast quantities of gas on-site, just enough to cope with daily/seasonal variability in demand, gasworks were a somewhat dangerous and definitely very smelly neighbours. Common by-products of almost all gasification processes are sulphur, ammonia and coal tar. Although all of these were stored and later re-sold (with some gasworks actually making ammonium sulphate fertiliser on-site) environmental friendliness wasn't really a priority and various leaks and spills were routine.

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As for how more rustic places were served... Again, it depended. Some villages and even individual families who could afford it had their own "gas houses" that were essentially scaled-down versions of the gasworks you'd see in cities. Acetylene and "air-gas" I mentioned earlier which require much simpler machinery to make were another option, and some varieties of oil-derived synthetic gas (most notably Blau and Pintcsch gas) were actually compressed until liquefied and transported in tanks and canisters much like you would natural gas, those were used primarily in trains and lighthouses. And yes, quite often it was as straightforward as a pipeline from the nearest large gasworks, especially in the 20th century. A place like Downton Abbey (ADD: in the early 20th century it's seen in the show) could have used any of those options to get its gas.

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If they did use coal gas it was a remote outbuilding (remember, those things stank of sulphur and ammonia) served either by a full-time operator or one of the gardeners. Air-gas system required a licence, and although it wasn't a hard requirement, the source below indicated that licences for such a system not in some kind of outbuilding were usually denied. So, some outbuilding in an area where neither smell nor fire risk are of concern is the most likely place if the gas wasn't simply piped from a big plant.

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If you need more details about country houses and other remote rich dwellings specifically, "Petrol Air-gas; a practical handbook on the installation and working of air-gas lighting systems for country houses" by Henry O'Connor is a 1909 British handbook available here: https://archive.org/details/petrolairgasprac00oconrich/page/10/mode/2up

It has a comparison of air gas with other systems commonly used in rich country houses at the time (carbide/acetylene, electric and conventional coal gas,) ads from various companies selling and installing gas lighting equipment, photographs and schematics, excerpts from safety and licencing regulations, and other kinds of practical details.

Here are photos and a short description of gasworks of a Scottish castle, these used coal gas and acetylene: https://canmore.org.uk/site/121401/culzean-castle-gas-house

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Not likely to be used in domestic lighting in all but very unusual circumstances, but here's an article from old Scientific American with a description, drawings and photographs of a Pintsch gas works and the use of Pintsch gas for train lighting:

http://www.catskillarchive.com/rrextra/sdgas.Html

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There was also a parallel industry of non-lighting gas use, most interestingly gas-powered internal combustion engines, You could run them on lighting-grade gas, and later gasworks often used them in place of a steam engine to power various equipment, but there were several processes that produced gas that was unsuitable for domestic use and used exclusively to power those engines ("producer gas.") Liquefied Blau gas was used as fuel for Zeppelins, and in the 20th century cars and tractors powered by onboard gas generators (which used mostly wood and a variant of complete gasification process) were used, they were especially popular during the WWII.