I would recommend using mailchimp. I know that there are many decisions how to receive emails, how to send them, but use the prestashop official module to receive a subscription. You can open the module newsletter and export clients to csv. You can import it into mailchimp. You can also install the mailchimp module to synchronize this process.
Mailchimp gives you the opportunity to view detailed statistics: who opens your newsletter, the most popular URL with a click s, successful sent email percent ant who didn t receive email. Here you can see all the data.