Presumably privacy.com knows your real identity, and presumably the privacy.com credit card number you give to the merchant can be traced back to privacy.com. From there the merchant is just a court order away from getting your contact info, and if you actually do owe them money, I don't think a judge would balk at that, at least not after the merchant has filed suit against you (as John/Jane Doe).

But of course no merchant is going to go through the effort and expense to do all this, unless you owe them a decent chunk of change. A couple months of a $15/mo subscription likely doesn't clear that bar.