I am having the same problem. But on one of my child portals (not all, just one) I've dug down and found some other odd behavior which is hard to explain but Ill do my best...
I made the following changes:
Page Settings have the following roles checked
- 'All Users'
- 'PASA Editor' - this is the new role that was created
Module Settings have the following checked
- Every role has View Module permissions
- PASA Editor has Edit Module permissions
Result - A user logged in with PASA Editor permissions does get a module drop down menu but the menu only contains the Help link.