Each chain forms an entirely separate storage structure – if an organization is connected to one chain, it doesn't get any idea what is happening on another. And the addresses and permissions for each chain need to be managed separately.
But within one chain, each stream is visible to everyone connected to the chain. Using MultiChain Enterprise it's possible to have read-controlled streams, but other nodes will still see the presence of stream activity.
You can think of it quite like the difference between different databases and different tables, in the SQL world.