With the admin user I can do everything, add, delete, modify, etc. As it should be.

Then using this user added a new one, with full access to sales and accounting options as some other ones needed for sales operations. Then tried to add a new supplier and it throws a warning:

Access Denied
The requested operation cannot be completed due to security restrictions. Please contact your system administrator.
(Document type: Pricelist, Operation: read)

Tested all the possible combinations on user rights and ended up giving full access to all options. Nevertheless I'm unable to create any new contact/customer/supplier/etc with this user. I'm running out of options, I don't know what else to test or where to look.

EDIT 1

We have 3 companies, no hierarchical structure, all are on the "root" of the organization. None of them is on selected on the "Default Sales Pricelist". For now we have only one warehouse for the stock of one of the companies. And the two default locations for Stock and Output.

EDIT 2

Just in case looking for the answer, it is on the comments of the selected answer. It is a reported bug. Go to multi company Access Denied Document type: Partner, Operation: read where the bug was reported and is followed up by other users.