There is no guarantee that a private lawyer will get a better deal. However, the biggest advantage a private lawyer has over a PD is that a private lawyer is in court to handle your case and your case alone, not an armload of cases that the PD's must maintain. You should be able to think over an offer for at least a short period of time. If you don't like what the PD negotiated, you are the one who would make the final decision on whether to accept the offer or take the case to another level or to trial. The attorney is there to advise you, based on their expertise & knowledge, but it is your decision on whether to accept & take the offer.