You should not worry about the list and "production data". The fact that the source list is not marked as "Production data" has nothing to do with this. Production Data relates to the way data is synchronized between Production and Development models through ALM.
Since neither the mapping itself (as the action works for the WS admin) nor the user permissions seem to be the issue, as you pointed out in your original post, I agree with @Misbah on the likelihood of "dynamic cell access" to be the reason.
Please, check the following: Can you read the data from the source module? Can you write in the target module?
If the answer to any of the previous questions is "no", please ask your WS admin to check the following:
1. In Users: check what is the Role assigned to your users. Then in User --> "Roles - Modules" tab, check if your role has the proper Write / Read access permissions for the source and target modules.
2. If the Role is fine, then check the Dynamic Cell Access: the WS admin will need to go to the module where you cannot read / write data, open the blue print and scroll right until he/she finds these columns:
If dynamic cell access is enabled, some of the cells in the columns show above will have a formula pointing to a "boolean" formatted line item. Check in which module that LI is and make sure that is not restricting your uses read / write access.