It really depends on what you are going to use those proxies for. If you are going to need the same IP for a longer period of time (for example Facebook account), then you will need a fixed or static residential IP.
If you are going to use those proxies for lets say scraping, then you will need rotating IPs, because this way you won't destroy the IP by sending too many requests from IT over a short period of time.
However, I don't really understand what you mean by "private or working". If you're asking, whether all residential IPs are assigned to one user, then the answer is no, as they are divided between dedicated (you are the only one to use them) and shared (plenty of people can get the same IP address). They've both got different use cases, but dedicated residential proxies are more expensive, so it is worth thinking twice, whether you actually need them. And if you're asking, whether you can feel safe, while browsing with a proxy, well, yes and no, as unlike vpns, your connection is not encrypted, but your original IP is still hidden.