Not a lawyer, this is not legal advice.
This is a contractual issue, more than anything else. However, a principal reason for using hosted payment providers is that they handle all the sensitive payment information, not you. If it is well built, you should never even see that payment information (most redirect completely to the other site and then redirect back to you so that they can be assured as possible they are under their own control - they prevent frame embedding and so forth). If the payment processor allows you to build the form, including the CC data, and call their backend, you will likely have more liability (because you have that access - you could do something dumb like logging the data before sending it to the processor. Never do this.)
If there is a compromise, you will likely be informed by your payment provider and be the one who contacts your customers, because you are the one they have a relationship with. Even this may not be true, however.
If you want to minimize liability, do everything payment related on their site - you should be only informed that the payment was made, not any payment details. Get your contract read by a lawyer and make sure it spells out liability issues and your responsibilities and make sure that is acceptable to you.