Great, so your organization is created and you've invited members of your team. How do you actually use the network? That really depends on your needs and what you're tying to accomplish.
Think of your organization as a shared workspace, an open office or an intranet. Everyone on your team who've you've invited can interact in the space in their own way while being aware of activity by the rest of the team - though not necessarily working the same project. Is that vague enough? Let me explain in more detail:
There are a few scenarios:
#1. Each member can spin up any type node they wish - Bitcoin, Stellar, Aion, Ethereum, and the nodes don't interact with any of the other nodes other than simply being on the same network. That's the open office plan, shared workspace approach
#2. Shared resource. An organization of multiple users can share access to a shared node, each accessing it for their own or team use. Think of a company which has a shared Ethereum node. The access information is available to everyone on your team to use.
#3. A private network (currently only Ethereum) for deploying smart contracts and other consensus projects. Complete details are in individual articles per node on smart contracts, but the short pitch, the basic principle is that each node across the network has a list of their peers and that is who they interact with. Each node does not need to know every node in the network, just a subset. If one of those nodes goes down or becomes deactivated, they always know of another node they can interact with