I am running K2 on two test sites and I am having some permission issues.

I think I have set the K2 permissions the same for both sites and have compared the settings to make sure I am not missing anything.

Basically, one account is missing the Add new item menu/tab for the Registered Users group whilst the other is not.

I will appreciate any insight into this problem.

Thanks

Make sure that you have enabled Frontend Editing for both sites.
Then check if your users have the same permissions and that they are assigned to the same group that has the access to perform Frontend Editing.