When you invite someone to an organization, they are not added to any initiatives or projects. They are simply added as a organization member.
When you invite someone to an initiative, they are added to ALL projects within that initiative. They will be set as workers on these projects.
If you want to invite people on specific projects, you must go at it one project at a time.
People will only see items in projects they are part of. They will see all the details of all the items within the projects they are part of.
For more information:
Inviting to Organizations
Inviting to Initiatives
Inviting to Projects