There is an immense range of situations. Schools run from small and friendly to large and businesslike. This diversity makes generalizing about their pay and work conditions very difficult. Significantly, they are the easiest venues for beginning teachers since many require no teaching experience and few credentials. Organizations may also directly engage a private teacher rather than contract one through a language school.
Private teaching is informal, flexible and pays a good hourly wage.
However, it is exceedingly difficult to find enough students to live solely on such work. Also, private students can be unsteady and it may be difficult to get a work visa without a school's sponsorship. Still, taking students on the side augments one's income and is a good way to make friends. On the most basic level, anyone wanting to be English teachers must be native English speakers or VERY close. Secondly, it is generally assumed that teachers have university degrees.