I have a multi site with 4 sub-sites. wpForo is set up on one of the sub-sites, while the others use the default WordPress user management system. The problem I found is that when I attempt to register on the forum with an email address that is already registered on one of the other sub-sites, it tells me that the user already exists even though they are not on the sub-site. This would suggest that the forum plugin is checking for users at network level. Interestingly, when I try to log in to the already existing network account on the forum, it ends up in a critical error since technically, the user does not exist on the forum system. Only at network level.
This will quickly become a problem as users who have an account on one of my other sub-sites will be unable to use the forum. If they try to register, it will tell them they already have an account - and when they try to login, it results in a critical error and the forum becomes unavailable to them until I log them out manually from the admin dashboard.
The way WordPress multisite works with user management is this - if I have 4 sub-sites, and a user is registered on only one of them, and they go to another sub-site and try to login with their information, WordPress automatically logs them in and gives them access only to the sub-site they actually have an account on. They cannot access the sub-sites they have no account on but it does not throw an error at them. It works and though somewhat confusing, it does not have errors.
So what I'm reporting with wpForo sounds like a bug to me. At the very least, it should be checking for existing forum members on sub-site level not on network level as this is causing issues. So, even though a user already exists on the network, they can still register with the same email address on the forum without issues - or at least be able to log in and then proceed to the forum without errors. I understand that the forum's user management uses the default WordPress user management system but this needs attention. The way it is now, users will be unable to use the forum if they commit the "sin" of having an account on another sub-site first.
your question is not related to wpForo, however if the user is registered to the network, and he cannot log in to the forum sub-site, that means that he needs access to that sub-site, so you need to give access to your users for the sub-site.