Christians teach that Jesus died on the cross for our sins and that his sacrifice is the only way to receive eternal forgiveness and life. Muslim leaders reject this idea. They say that even if Jesus did die on the cross it is not possible for him to bear our sin because the Qur'an says,
Every soul earns only to its own account; no soul laden bears the load of another.
(Qur'an, 6:164, 17:15, 53:38, Arberry)
However, it seems that Islamic leaders only show some of what the Qur'an and Hadith actually say about this subject. The Qur'an actually has verses which say that one person can bear the sins of another.
That they may bear their loads complete on the Day of Resurrection, and some of the loads of those that they lead astray without any knowledge. O evil the load they bear! (Qur'an 16:25, Arberry)
They shall certainly carry their loads, and other loads along with their loads, and upon the Day of Resurrection they shall surely be questioned concerning that they were forging. (Qur'an 29:13, Arberry)
Also in the Hadith we see that a person can bear someone else's load.
Narrated Ibn Abbas: A man came to the Prophet and said, "O Allah's Apostle! My mother died and she ought to have fasted one month (for her missed Ramadan). Shall I fast on her behalf?" The Prophet replied in the affirmative and said, "Allah's debts have more right to be paid." ... (Sahih al-Bukhuri: vol. 3, bk. 31, no. 174, Khan)
Narrated Ibn 'Abbas: A woman from the tribe of Juhaina came to the Prophet and said, "My mother had vowed to perform Hajj but she died before performing it. May I perform Hajj on my mother's behalf?" The Prophet replied, "Perform Hajj on her behalf. Had there been a debt on your mother, would you have paid it or not? So, pay Allah's debt as He has more right to be paid." (Sahih al-Bukhari: vol. 3, bk. 29, no. 77, Khan)
Narrated Ibn 'Abbas: The mother of Sad bin 'Ubada died in his absence. He said, "O Allah's Apostle! My mother died in my absence; will it be of any benefit for her if I give Sadaqa on her behalf?" The Prophet said, "Yes," Sad said, "I make you a witness that I gave my garden called Al Makhraf in charity on her behalf."
(Sahih al-Bukhari: vol. 4, bk. 51, no. 19, Khan)
When we consider the whole picture of what the Qur'an and Hadith say we see that bearing the load of another is a part of their teaching. I am not saying it is exactly the same as Christianity but the idea is there.
The next time a Muslim says to you, "each can only bear his own load", rather than go straight to the Bible for your answer, first show them that there is a bit more to what the Qur'an and Hadith actually say, then go to the Bible.