We recently ran into a problem on an implementation of Sharepoint 2007 where the 'Personalize this page' would not show up for the root page of Enterprise Portal template. Since this caused a lot of headache I wanted to post a few comments on it and what we did to fix it.
1st, we verified that all permissions were setup and that the personalize this page feature was turned on within the MOSS Central Admin.
2nd, after digging into it we found that default.aspx was actually checked out to one of the admins, after checking this in we did start seeing the 'personalize this page' within the menu when you click on the name of the user signed in.
The problem still existed where the user could add new web parts but they could not move web parts around. Since we verified that within each web part they should be able to (i.e. 'Allow Close, Allow Hide within the web part properties) we still were not sure. We ended up finding that there was a flag within each 'Zone' setup in the default.aspx page. We opened up the file in Sharepoint Designer and modified the property to 'AllowPersonalization' and then it started working. I just wanted to add a post to the forum in case other users run into this.
-Chad