Omnipay doesn't support notifications for existing payments. It automatically handles the first notification which is made directly after a payment, but doesn't yet have methods to handle notifications if you change the status of a payment (for example by refunding it from PayPal's control panel, it will not update in your database automatically).
Note this isn't needed for the initial payment. Especially with PayPal Express the payment is not confirmed until the customer returns to your site, so there is no chance of them closing the browser window and the payment being lost.
If you want to set up a notification endpoint and listen for events which happen after the payment is processed, you will need to add a new request type to Omnipay.