My father died in a car accident and the driver of the other car also died. The traffic report determined that each of them was 50% responsible. What do we, his sons, have to do?.
If it has been determined that both of them were equally at fault, then each of them has to pay compensation for damage caused to the other, whether it was loss of life or property, according to the degree of his error.
As the traffic report determined that the share of error of each party was 50%, then each of them must pay half the diyah of the other, and half of the cost of the damage caused to the car.
The diyah in this case must be paid by the male relatives of each of them, meaning the male relatives on the father's side, such as brothers, paternal uncles and their sons (i.e., paternal cousins).
With regard to expiation, each of them must offer expiation in full for killing by mistake, because expiation cannot be divided.
It says in Fataawa al-Lajnah al-Daa’imah (21/273):
If the accident was the fault of both drivers, then each of them who contributed to killing must pay a share of the diyah, commensurate with his portion of blame. With regard to expiation, each person who played a part in the accidental killing must offer expiation, even if his degree of blame is less. But if his share of the error is 100%, then he alone -- and no one else -- must offer expiation and his male relatives on the father’s side must pay the diyah. End quote.
If the heirs of each of them agree to waive their rights in return for the other party also waiving their rights, there is nothing wrong with that.
The expiation for accidental killing is to free a believing slave. If that is not possible, then it is to fast for two consecutive months.
As your father died in the accident, if it is possible to find a believing slave, then you must take money out of his estate to buy this slave and then set him free.
If it is not possible to find a believing slave, then you do not have to fast on his behalf, but if one of you volunteers to fast on his behalf, that is good and beneficial in sha Allaah. It is not permissible for more than one person to take part in the fasting; rather the two consecutive months must be fasted by one person.
It says in Fataawa al-Lajnah al-Daa’imah (21/324): If your father's situation is as you describe, then he has to offer expiation for accidental killing, which is to free a believing slave. If that is not possible, then he must fast for two consecutive months, because Allaah says (interpretation of the meaning):
“It is not for a believer to kill a believer except (that it be) by mistake; and whosoever kills a believer by mistake, (it is ordained that) he must set free a believing slave and a compensation (blood money, i.e. Diya) be given to the deceased’s family unless they remit it. ... And whoso finds this (the penance of freeing a slave) beyond his means, he must fast for two consecutive months in order to seek repentance from Allaah. And Allaah is Ever All‑Knowing, All‑Wise”
As he has died and did not do what was required of him, then his next of kin has to buy a believing slave out of the estate and set him free. If there is not enough money in the estate to do that, or if there are no slaves, then it is mustahabb for his next of kin to fast on his behalf for two consecutive months, because the Prophet (blessings and peace of Allaah be upon him) said: “If a person dies owing a fast, his next of kin may fast on his behalf.” Saheeh – agreed upon. End quote.
It also says (21/326): It is not permissible for more than one person to share the expiatory fast; rather what is prescribed is that one expiation should be undertaken by a single person. End quote.
And Allaah knows best.