"The employer" can certify workers to enter permit required confined spaces. OSHA does not list any specific requirements for who the trainer can be, the training itself just has to meet the requirements of the regulation.
29 CFR 1910.146 mentions the training requirements clearly:
"The employer shall provide training so that all employees whose work is regulated by this section acquire the understanding, knowledge, and skills necessary for the safe performance of the duties assigned under this section."
The regulation also mentions that the employer must certify the training has been accomplished.
"The employer shall certify that the training required by paragraphs (g)(1) through (g)(3) of this section has been accomplished. The certification shall contain each employee's name, the signatures or initials of the trainers, and the dates of training. The certification shall be available for inspection by employees and their authorized representatives."
It is still important to designate a trainer who has the knowledge, training, and experience to certify confined space workers. It is up to the employer to determine if the training provided was sufficient.