I agree with the rejection by u/mimicofmodes of your claim that newspapers were not a common place for advertisements offering or seeking employment in household service. If you look at various memoirs and compilations of oral histories of servants in late 19th century Britain, for example, you will see that many found their employment through newspaper advertisements.

As noted in the linked response to the question about maids' uniforms, the importance of word-of-mouth recommendations (including written correspondence) can't be overlooked. This was especially important in the period before the railroads made travel easier. Travel then took much longer and was more expensive, meaning it was much less likely lower class people would travel from, say, York to London to work as a domestic servant. Those working in service would generally seek employment locally, finding it through a personal connection or perhaps in a regional newspaper.

With the advent of the railroads, national newspapers like the London Times became more important not only for employers seeking servants but for servants seeking an employer. According to Michelle Higgs, the Times promoted itself to servants seeking employment as an upscale publication with a wealthier class of readership able to provide better living conditions and pay. Employers, on the other hand, were able to specify things like religion, age, and even height when placing a newspaper advertisement. They also occasionally posted listings seeking employment for their current servants if they could no longer keep them on because of a change of circumstances, such as children outgrowing the need for a nanny.

Higgs also mentions the establishment of servants' registers, which were basically employment agencies in larger cities. Servants were able to sign up with these services, providing information about their personal and professional background and listing their own requirements for employment. This type of arrangement, however, was susceptible to fraud, and so many servants shunned it.

Finally, the growth of illustrated periodicals provided an additional opportunity to seek employment. Women's magazines were a common place for people to offer their services or for employers to advertise open positions in their households. The most famous of these (and one that continues to be important for the domestic service today) is The Lady, founded in London in 1885.

SOURCE:

Higgs, Michelle. Servants’ Stories : Life Below Stairs In Their Own Words, 1800–1950. Barnsley, England: Pen & Sword Military, 2015.

There's not a good way to answer this, because in the nineteenth and twentieth centuries, people did make frequent use of newspaper ads for hiring servants. I discuss this to some extent in both a follow up to a question on French maid outfits and How widespread was anti-Irish sentiment in the United States in the late 1800s-mid 1900s, particularly the infamous "No Irish Need Apply" signs? However, as stated in the first link, people also made great use of employment agencies and word of mouth in hiring.