Almost everyone would look forward to having his or her judgment recovered on a pure contingency basis. It implies that the collection lawyer in NY would have to advance his time and expenses without charging anything until the judgment is recovered. In a pure contingency basis agreement, the lawyer would get his expenses repaid to him first before splitting the recovered money with the judgment creditor client. With a smaller and riskier asset of the judgment debtor, the chances of the lawyer recovering a judgment on a pure contingency basis would be relatively less. 

You could negotiate several aspects. It would depend on the details of the situation. Most judgments might be strong enough that a contingency collection lawyer might recover the judgment on a contingency basis. On other judgments, the lawyer would need his client to pay some or all the expenses. It would not be wrong to suggest that the time of the lawyer would usually be reimbursed on a contingency basis. Most judgments would not have adequate debtor assets for the judgment collection attempt to be performed on a contingency basis. In such a situation, you should pay the retainer, by the hour, and pay all expenses. You could also find a regular contingency judgment enforcer. 

Yet another option would be to try to sell your judgment for cash upfront. However, if the lawyer would not recover a judgment on a contingency basis, the judgment would probably be worth every dime paid upfront. Contingency rates would vary, based on the strength of the assets of the debtor. A large judgment against a rich debtor would have a contingency rate of 30% for the lawyer and 70% to the judgment creditor. A riskier looking judgment might be 70% to the lawyer and 30% to the judgment creditor. On the same judgment, a lawyer would absorb all expenses and charge 50% on a contingency basis while others might charge 25%, and the client would pay the remaining amount.