Moved to professional services group.

Thanks.

Though,  imho, a part of the question is highly technical and could have evoked many interesting technical propositions or clues or even solutions!

@dranii You can do it with WebServices and JSON/cURL

I would love to see this developed, I would be happy to help where I can!

You can try https://github.com/mapkyca/Elgg-Multisite (1.8 only)

Cross site login can be acheived with Simple Saml https://github.com/ColdTrick/simplesaml

They can be combined, but it is quite difficult to get this up and running.

add friends on another domain and show them in his profile widget, friend list etc

A multisite (both sites in same DB) solution would be the most reasonable but still require core changes to allow displaying entities from a different site, and only when you want that.

Simpler but still tough would be single sign-on between the sites and each site can have proxy objects for the other site. The proxy would have to copy the title/name and maybe description, but its getUrl() would just point to the real entity's URL on the other site, probably same for icon URLs. The proxy would probably have to have the same type/subtype as the original. Getting this working just for users to start with would be cool. And of course, there are sync issues, and you'd have to hijack a lot of functionality like menus so you never offer to, say, add a proxy as a friend. Maybe no hover menus on the proxies.

This can be done with a plugin installed on both sites trying to talk or connect to one another.

I had done my own project several months ago. However, the project was on elgg 1.8 and Openx now known as Arivex adserver. https://community.elgg.org/plugins/1609647

The concept was, once a use is logged in elgg site and they want to publish an ad similar to Facebook ads service - The user will click a button upload ad. And within seconds, account will be created on another website within or not on the same database nor Web hosting company without the user leaving or logging out of elgg site. So long as the user is logged in elgg site, their ssession on the other site is active. Therefore, the user can pull or interact with the other elgg site without any headache and even not realising that he or she is interacting with two elgg site without pulling the hair or pointing the fingers at the computer screen. Once the user is logged out of the current site, the plugin automatically logges the user off from the other site.

The only problem will be, the elgg site accepting other sites to pull or push data from the site, will first of all be assured that the other sites are not accepting spammers on their site.
I posted this comment via cellphone while driving home. If my comment is not clear feel free to give your feedback.

http://friendica.com/   https://gnu.io/social/   https://github.com/redmatrix/redmatrix diaspora and to some extent idno/known probably, does this. Its woth seeing/studying/analysing what these do and what exact feature these have imbibed.

What makes the RedMatrix unique is what we call "magic authentication" - which is based on our groundbreaking work in decentralised identity services. No other platform provides this ability. Within the matrix the boundaries between different hubs are blurred or seemingly non-existent. Identity in the matrix is considered transient and potentially nomadic. "Who you are" has nothing to do with "what computer you're connected to", and website content can adapt itself according to who is viewing it. You have the ability to "clone" your identity to other hubs; which allows you to continue to communicate with your friends seamlessly if your primary hub is ever disabled (temporarily or permanently).

The above is what Red says, however what I feel none is as easy to install and test as Elgg so if Elgg is going to do this 'federation' it will be a massive gain, though we need to define first what are the things we need to do. Whatever, the time is more than ripe for this now.