The simplest way to explain how this works is that the Newsletter components (popup and either a newsletter section or the footer) of the storefront will send email addresses entered by shoppers over to the Shopify Customers list - this much is controlled by the theme.
The Shopify customers list is outlined further here:
This is controlled by the theme.
After this, any installed third-party app, such as Klaviyo takes over and should connect up to your respective external account. The best way to check things would be to add a test email address into the sign-up field on your storefront, then check it is added to your Customers list in the Shopify admin.
If this works without issue, then the theme is doing its job. If the email entered does not then appear on the third party app's side, it is a connectivity issue with the app being used and you would need to reach out to the app developer for further guidance.
Each Shopify App should have their respective Support Details listed, on their respective page of the Shopify App Store, should you find you need their assistance - to use Klaviyo again, as our example:
Here are three apps which are popular with merchants: