Legal obligations do not appear out of thin air. You have no direct contract with Visa or MasterCard or any other card association, so they have no power to require you meet any of their standards. It's your contract with the specific processor that must have a clause requiring you be compliant or requiring you meet all requirements of the Visa and MasterCard Operating Guidelines that creates this responsibility. So it is completely dependent on what back end processor you contract with, in the sense of that's who creates your legal obligation and the party that has the power to enforce the contract if you do not meet that obligation.
In this case, it is the "PayPal Website Payments Pro and Virtual Terminal Agreement", that you must agree to as part of signing up for that service, that has the clause "Compliance with Data Security Standards" where you agree to be compliant.
The relevant portion for the OP's question about when, if ever, they require proof of compliance:
I didn't notice any other mention of showing proof of compliance, but I was only skimming.