If you are using a staging site that is cloned from your live site, there could be some additional issues with emails that are being sent to your customers from the staging site.
Most often, issues here are related to Reminders emails. If you have Reminders configured on your Live site, the reminders will also be set up on your staging site. The staging site will send out the reminder as well as the live site so the user will receive duplicates.
To prevent this situation and be sure that this will not be an issue on your site, you can install the Stop Emails plugin. Having this plugin installed and activated will stop any outgoing emails generated by WordPress core, plugins, and themes.
Otherwise, you can simply delete any created reminder emails from your staging site to ensure they aren't sent out either as duplicates or when they shouldn't be sent out.
Additionally to all mentioned above, you will need to Disable auto-responder integrations (MailChimp, ConvertKit, etc.) if you have any.
When this option is checked within MemberPress, and the membership has its own auto-responder list set up under its Advanced tab, the global list you set on this page will be disabled on that membership page. The member will only be added to the membership's list and not the global list in this case.
Finally, since WordPress Cron is used for sending most of the emails from your site, make sure to disable WP-Cron as well. For disabling Cron jobs on your WordPress site, you will need to add a line of code into your wp-config configuration file before the line ‘That's it, stop editing.'
Disable WP Cron – define(‘DISABLE_WP_CRON', true);