The answer accepted above is correct if the intent is for you to have a private copy of the notebook, but not if the intent is for the notebook to be shared through SharePoint 2010. In the latter case, you should not copy or download it locally, or even check the notebook out except to deliberately disable co-authoring.
Instead, open the notebook through either:
- The link on the SharePoint document library page
- The link that you can send by email from that page
- A bookmark you have created to that link
I am currently researching how to use a shared OneNote notebook inside SharePoint 2010 for a project at work, so I will share what I have found so far. This is a good opportunity to organize my research (my co-workers will need the same information), so this post will also be a draft of my research report.
Microsoft calls this kind of sharing "co-authoring." It can be done quite easily but requires some initial setup. As best I can determine, if all your users have OneNote, all you need to do is set up a document library and grant the users access, then create a new shared notebook and specify that document library as the location in which to place the notebook. If some users don't have OneNote installed, you can have your SharePoint admins install Microsoft Office Web Apps on the SharePoint 2010 server. This allows all users to view and edit the notebook from their browser window.
Storing the shared notebook inside SharePoint 2010 takes advantage of the best of SharePoint and the best of OneNote. (Optional: Apparently you can store the notebook inside a Skydrive folder and get the same effect.) Copying the file locally loses all the benefits of storing the shared notebook inside SharePoint -- even if you enable sharing on your local copy of the notebook. In that case you would only get OneNote's native benefits of sharing.
This article describes Microsoft's various approaches to co-authoring in SharePoint, including OneNote:
Document collaboration and co-authoring:
These articles describe how to use a shared notebook inside SharePoint 2010:
Collaborative Note-Taking with SharePoint 2010 and Microsoft OneNote
Microsoft SharePoint 2010 Conference Live: Co-Authoring in Office 2010
Tip: You can prevent co-authoring a notebook that is shared through SharePoint by checking out the notebook. Others can view the notebook but cannot change it until you check the notebook back in.
You can find more information on using and setting up OneNote notebooks within SharePoint 2010 at these Technet links:
Co-authoring overview (SharePoint Server 2010)
Co-authoring administration (SharePoint Server 2012)
I found several other related links but several are descriptions of presentations that lead to broken links and these links here (working at press time in April 2012) seem to cover the most important information for getting your co-authoring solution rolling.