Interesting, do you have some examples of this, I was under the impression that you wouldn’t be allowed to keep some of the company private and some of it public at the same time. I’ve tried Google and it’s difficult to find information about this, the Wikipedia for IPO doesn’t mention this either.
You can look at individual IPOs. The most high profile one recently was Airbnb. According to the NYT [0], they sold $3.5 billion worth of shares, at a valuation of $47b, so only 6%. For another example, Mark Zuckerberg still owns 30% of Facebook, 8 years on from the IPO.
You're correct that the whole company "goes public" at the same time. But that just means it becomes legal to sell shares to small investors, and the company becomes subject to the reporting requirements, etc. The previous owners still typically own most of the shares, and (almost always) are forbidden to sell them for around 6 months after the IPO.
The full company goes public, however existing shareholders keep their shares and don't have to sell them - they now just have shares in the public company.
This is why dilution happens too - new shares are created so that they can be sold.
