When a customer subscribes to your Messenger Campaigns using the Messenger Widget, they will become subscribers for 24 hours during which you can send them Abandoned Cart Campaigns with promotional content. However your Facebook page won't have any access to their public info.
On the other hand they can become permanent subscribers if they subscribed via the Welcome Popup or the Messenger Add to Cart Popup, or write to your Facebook page (this includes replying to your Messenger Campaigns).
These permanent subscribers can be seen on your Subscribers page.
When you send a Sponsored message, it can only be delivered to your permanent subscribers.
The best tactic you can do to grow your subscriber base is setting up a Welcome campaign for your subscription tools. You can use this to send an automated message the moment people subscribe. It's a good way to provide a warm welcome to them, and make them reply back to your site.