There should be a checkbox when you create or edit the the organization to share users and also allow it to be a subscriber or to container products,libraries,or projects.
You can set the organization to whichever they are principally associated while creating user, but define them at site and do not select domains. This should take care of things. To give everyone at least view/read access, add the organization to the guest role on a team. I do not like the guest or members roles, so I define my own to retain full control of system, but that may be too advanced for your requirements.
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