When clicking on document, not prompting with 'Read-only' or 'Edit'

The read only or edit dialogue box shows up only if the document library has check in-out activated ( = turned on).

Other than that, you should get your IT personnel to perform a repair on Office.

If the issue is related to specific people, you should make sure that office 2010 x86 has been installed on their machine , also Sharepoint designer will solve this issue.

Also you should make sure that Default open behavior for browser-enabled documents has been checked in library setting

Go to document library's Settings, > click Advanced Settings.> Find the option for "Default open behavior for browser-enabled documents", and select "Open in the client application".

Now, when a user clicks on the document name to open it, they will be able to see that they can edit it and select the "edit" option from the prompt (assuming you are using IE). Here is what that should look like:

Ref