Ideally I should be able to share an individual task or a single branch of tasks inthe outline with a person. When I add the +John in the task description, this task and it's sub tasks should be shared automatically with John.

Since this doesnt'seem to be the case and the sharing level is established at the whole file level, then it's logical that you should be able to have 2 or more files opened simultaneously.

The use case is that you have
A) Personal Outlines (your own personal task and projects)
B) Family Outlines (family projects in which several family members share and participate)
C) Job Projects (this people should definitely not be able to see the other groups). Inside you might be a team member in different projects which involve different teams and whose task outlines shouldn't be mixed.
D) Entrepreneur Projects (definitely not to be seen by the Job team members)

Because obviously these identifies different groups of people with their own visibility, it definitely is clear that being able to see all your tasks from all your projects together is desirabel without needing to switch between the groups described above constantly.

So either enable the sharing at the task level, or enable having several files open simultaneously

The document name should also be in the window title also. So the user can tell the different instances apart.

This is also important if one wants to share different tasks with different people. For example, I want to have a personal document that I don't share and a project document that I share with collaborators, but see all my todos at once...

+3 - yes please!! I need one for personal and one for professional and I need both open at all times!! Hope this gets implemented fairly soon - thanks :-)

As a sample use case: I want to use Moo.do for my detailed task lists for specific projects and for ongoing notes, but I don't want to combine these into a single document.

It's kind of surprising Moo.do doesn't support this already. You can open two pages on Moo.do, but if you change the document in one you are forced to reload and change the document in the other one.

As @KV mentioned, the URL fragment should update wth the document name also so we can direct link to it. And, of course, the window title should change to match the current document.

I've thought about trying http://fluidapp.com/ to work around the single document limitation, but it really feels like Moo.do should be able to solve it natively. Thanks!

+++
"and the ability to decide which of the documents (or all) appear on agenda pane".

Totally agree.Thanks.

Yes, and the ability to decide which of the documents (or all) appear on agenda pane.

Yes, very necessary or at least much much faster switching between documents.
Plus 1 on the ability to direct link to a specific document.

Plus the ability to direct link to a specific document

Yes Please!! Very important!

I cannot mix drafting with my essential tasks. Not even in focused view.