This thread is resolved. Here is a description of the problem and solution.
Problem: Which WordPress role I need to assign to a user to be a translator? I tried different roles and only administrator role can accept translation jobs so that I think I need some settings or something doesn't work correct.
Solution: Each user role has some capabilities. You can see here https://wpml.org/documentation/support/wpml-admin-capabilities/ a list of all the capabilities and the changes that can be done by the user roles that have them.
As you can see, in order to be able to add translations, the wpml_manage_translation_management capability is needed and the administrator role is the only one that has it.
If you want to let other types of user roles be able to add translations, you need to edit their capabilities. You can do this by using some custom coding or you can install a plugin that adds this option.
Relevant Documentation: https://wpml.org/documentation/support/wpml-admin-capabilities/
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