I couldn't remember any more details, so in the interest of science I've just done the experiment again.
I have transferred a .com domain to DH. In the DH registration panel, the box which says "keep my registration details private" was pre-checked by default and I left it unchanged. I only changed the email address which was pre-selected in the form.
On the plus side, the process went smoothly and took 32 minutes from when I credited my DH account with $11.95 from Paypal, until the confirmation email that the transfer was completed (and would have been quicker if I hadn't been doing other things at the same time).
Immediately after getting that confirmation email I tried a lookup of the domain on whois.sc (which is a front-end for http://whois.domaintools.com/) and the following thing happened, even stranger than what I mentioned in my previous note:
as the details were filling in (on my not-very-fast internet connection here) I glimpsed the phrases "happy dreamhost" and "private registrant", but then within a couple of seconds the screen auto-refreshed in front of my eyes and then showed my actual private registration details.
I then tried a few more whois websites and they are all showing my private registration details.
EXCEPT Dreamhost's very own whois website, at whoisweb.dreamhost.com, which is forlornly saying "Private Registrant" and "A Happy DreamHost Customer"
That auto-refresh at domaintools.com suggests that it is first gets some details from the registrar (Dreamhost), displays these while it is drilling down further, and then displays the raw info that it has drilled down to.
N.B. all this is specific to the case of transferring a domaim name in to DH. Based on my previous experience, if you then go back in to the Dreamhost panel and change the registration details, again selecting "private", then eventually your registration will be private. Soon I will do this. But right now, the situation is that
whoisweb.dreamhost.com is saying "Private Registrant"
every other whois website that I've tried is showing Real Name