http://dl.dropbox.com/u/49702772/Selection_062.jpg

I added those full permissions for Administrator group but that did not help. I'm not sure if it's a good idea to have everyone full access.

Sami, this type of error correspond to Access rules, For ex: There is a access rule defined on mail.message object with domain(filter) that logged in user can access only those messages which are create by him, so domain would be like [('create_uid','=',user.id)] and we can define operatons(CRUD) to be applied on this domain filter. Suppose we have defined all four (Read/Write/Delete/Create) rights, Now if anyhow user tries to Read/Write/Delete/Create any message which is not created by him, he will get exactly the same message as u got :
Odoo Warning - Access Error
The requested operation cannot be completed due to security restrictions. Please contact your system administrator.
(Document type: mail.message, Operation: read)
So, at your access rule(defined path above) please check for any such rules existance. and try disabling it.

It seems that even though I have added Administrator group full permissions to mail.message.all the default mail.message.all without any group defined over-rules the administrator groups rule. Or maybe it's the "Outgoing Mails" rule that needs to be changed/added? What would be the result if I added full rights to the default/non-group settings?

Thank you for your response, Temur. Your solution seems overly complex to allow admin user to send email invitation.

Actually I was focused on explaining that this is an expected behavior, and I answered the two questions in your post... I'll summarize my answer as: cloning process does not fail, but that's possibly an expected behavior. regarding your error message, it should be a solution to edit permissions on mail.message model from settings and allow your cloned user to read this model. you tried it? or checked if it was allowed to read mail.message model?

I think it can be solved by editing permissions of "mail.message" model at "/Settings/Technical/Database Structure/Models" page

And there should be more options to give this user read/write access to mail.message model at Settings page. As I explained, it's quite possible that firs user has permission because of it's id, but you can enable the copied user as well to have access to mail.message, by explicitly granting such permission to it's own group or to it, etc... there should be multiple solutions

Administrator group all ready has full rights to mail.message model. There should be no need to edit it or Administrator group.

... Can you update your question with full error message?

Here is the exact error message... Odoo Warning - Access Error The requested operation cannot be completed due to security restrictions. Please contact your system administrator. (Document type: mail.message, Operation: read)