Microsoft has decided that documents opened from network drives or email attachments are inherently evil, and so the default in Office 2010 is to open them in read-only mode. This gets irritating pretty quickly.
There are, however, many settings available to chance this behaviour, and I attempted, when setting up the installation directory for Office 2010 to preset them in suitable ways, such as specifying the V: drive as a safe source. However, I missed a couple of the possibilities - but you can make additional changes yourself.
The place to make changes in the security settings is called the Trust Centre. To get to it, select the coloured File tab on the ribbon, and click Options near the bottom of the left-hand column. The Trust Centre is the bottom selection in the Options window, and you then need to click the button labelled Trust Centre Settings...
Note that the settings available vary for the different Office programs, and (as described below) the setting governing the behaviour of Word when opened from Outlook is in the Word Trust settings, not the Outlook ones.
In the Trust Centre there is a wide range of possible settings, arranged in groups selected by clicking in the left-hand column menu, too many for me to attempt to describe here.
The one that several people have pointed out that I missed is in the Trust Centre for Word, under the menu heading of Protected View. In these settings, untick Enable Protected View for Outlook Attachments to allow editing and saving of documents opened directly from Outlook. Note that this setting is not visible in the Outlook Trust Centre.