In many ways, you might get a more pleasant working environment in private. Ie you get smaller classes and less disruption. Facilities are also good. In return you probably need to be there for longer hours in term time (although term times are longer) and there is a high expectation of wider involvement. Staff are expected to be fully committed beyond classroom teaching.
Private schools differ widely. Some are well funded with fab facilities, good fee discounts, high wages,plenty of on going investment and small classes,making with opportunities for promotion. These tend to be the more prosperous,larger schools.
Others are clinging on, close to closing down, especially some prep and pre-preps. The facilities can be outdated with no signs of investment for many years, no or very limited fee discounts, and poor salaries. Some have peculiar owners with no sense of current educational thinking, or no real plan for the future.
Look out especially for the very small school that has falling numbers, requiring year groups to be combined and people to teach far beyond their area of expertise. Look out too, for those with no affiliation to a bigger group of schools. Ask about 5 year plans for building work and other investment....quite telling about the financial position of a school.
So private can be a great place to work, or it can be worse and more insecure that state. Make sure you understand the sector fully and know exactly what kind of school you are considering working in.