The landscape changes fairly quickly. Indeed, there is a private sale that occurs before the pre-sale for large projects (who want to gather tens of millions for ICO, it's almost a must). By the time I am writing this post on April 2 - 84% of 2018 ICO funding came from private sales.
Usually it is open to crypto funds, VS or angel investors. For those seeking American investors, private sales comply with securities laws enforced by SEC - specifically Reg D, Rule 506 (c). It allows ICO projects to avoid the burdensome registration requirements, though it allows only the accredited investors to buy tokens ($1m + capital OR $200k+ annual income OR $300k+ annual income for a couple). That should not be a problem, since most investments in private sales fall within the 100 - 1000 ETH range.
Potentially, the SEC may go after these ICOs for not looking into the secondary trading issue, but that's a future and a US-based concern entirely. Many of the investors participating in private sales are, in fact, coming from Asia. Still, any ICO project considering doing a private sale is encouraged to look further into an article by Harbor "Introducing the Private ICO" on Medium, which analyzes various legal complexities.
Private sales often last 2 - 3 weeks, have minimum investments, and allocate 30 - 45% of the tokens as a cap. Unsold tokens are then sold during the actual pre-sale and crowdsale. The discounts offered during private sales are often steeper than those during pre-sales.