BuddyPress.org

I guess that means that when a user registers on one of the multi-sites he can use his account on all the other websites of the Multi-site installation, right ?

Yes, but understand that you have to give that user access to those other sites (in wp-admin) to write blog posts, for example. If a user does not have explicit permissions to a specific multisite site, they won’t have any special access; they’ll only be able to read the content as if they were another normal visitor. You can test this right now, without BuddyPress, and see how it works.

meaning it takes him out of the French Website to put him onto the English on

You seem to be using multisite to have your content localised into different languages. This is a common approach for WordPress multisite, but BuddyPress does not work the way you want it to in this situation.

You can activate BuddyPress in a few different ways. Simply, on multisite, you can activate the plugin on a specific site, or network-wide. There is lots more detail at https://codex.buddypress.org/getting-started/installation-in-wordpress-multisite/ but if BuddyPress is activated network-wide, all the BuddyPress content will only appear on one site (we call this the “root blog”). We use redirects to support this, exactly as you’ve found.

You can’t have BuddyPress active on a multisite that uses the sub-sites as different languages, and have the BuddyPress social network appear on all of them and work in the same way, except with different translations. It doesn’t work that way out the box…

…however here is an advanced setting you can use to make BuddyPress behave like this: BP_ENABLE_MULTIBLOG

See https://codex.buddypress.org/getting-started/customizing/bp_enable_multiblog/
I would encourage you fully test this setting out if you enable it, before you start building the rest of your site, just to understand and confirm its behaviour.

🙂

I think with multisite, WordPress user registrations only occur on the main/root site. I don’t know how BP_ENABLE_MULTIBLOG works with this (i.e. I can’t remember if the registration redirect happens). It’s been a while since I’ve used it.

If you’re a little technical, you could test it, and then look in the database afterwards — probably the wp_x_users, wp_x_usermeta, wp_x_signups tables, and see if there’s any field such as a blog ID or blog URL set somewhere for the new users that you could use.

Otherwise, I think you’d have to code something up.