This article https://signal.org/blog/building-faster-oram/ has some details but is more focused on improving their solution other blogs from the are "we want to build this soon" kind of blogs. It seems that most articles about this topic either have too little content to be of interest or are technology previews/"we maybe will do that" articles about things Signal wants to implement, where it's unclear if they did do that or something similar.

To cut it short they use Intel SGX to create a "trusted environment" (trusted by the app/user) in which the run the contact discovery.

In that trusted environment you then run algorithms similar to other messengers (i.e. you still need to rate limit them as it's possible to iterate _all_ phone numbers which exist).

If working as intended, this is better then what alternatives provide as it doesn't just protect phone numbers from 3rd parties but also from the data center operator and to some degree even signal itself.

But it's not perfect. You can use side channel attacks against Intel SGX and Signal most likely can sneak in ways for them to access things by changing the code, sure people might find this but it's still viable.

In the end what matters is driving up the cost of attacks to a point where they aren't worth in all cases (as in either not worth in general or in there being easier attack vectors e.g. against your phone which also gives them what they want, either way it should be suited for systematic mass surveillance of everyone or even just sub groups like politicians, journalists and similar).

https://signal.org/blog/private-contact-discovery/

I believe that the search term you can look for is constant time equality.