I just stumbled on a website called "WeAreTeacher Finder" that promises to link prospective students with language teachers for private lessons, either in person or online. The company's name is Teacher Finder. I have not used this service and cannot vouch for the website, but the best I can tell, the student-to-be does not pay for being connected to a potential teacher. I presume the teachers pay for referrals received, but without going through the process of signing up, I cannot verify that.
What has me intrigued is that--as one might expect/hope--the hourly rates displayed on the website are substantially lower than one typically pays when taking private lessons at a language school. From experience I know that it's not necessarily easy to sit at home in the US, bang away at a computer keyboard, and find an independent teacher who isn't going to have to turn over half of what you pay him to a language school (which really runs up the cost). A resident seeking to improve language skills would probably pretty quickly get the necessary information through the local grapevine or on a bulletin board somewhere. For a short-term visitor, it's not so easy.
For many major western European cities, Teacher Finder cites a cost for in-person lessons of 25 euros per hour. The hourly rate is somewhat lower in less-expensive cities like Budapest and Bucharest. It seems to be 25 GBP in England. In Switzerland--well, what would you expect? It's CHF 80-100!
In most cases the displayed rate is the same for any language offered by the enrolled teachers in that city (subject to the availability of teachers, of course, and I can well imagine that sometimes the cost turns out to be somewhat higher than that posted on the website), but in a few cases learning the local language costs more than learning a foreign language. I guess that's because of greater demand.
If by chance someone has had experience finding a teacher through this website, I'd love to hear about how it worked out. Teacher Finder is run by Andrew Davison, who is identified as "an entrepreneur and blogger" in Budapest. He wrote a post about Budapest markets that I encountered on the Budapest Local website, and there was a reference to Teacher Finder in tiny print at the bottom of the post.