وَلَا تَزِرُ وَازِرَةٌ وِزْرَ أُخْرَىٰ
The root و-ز-ر and this principle were analysed in detail in al-Najm at 53/38 (the weight carried on the back; wazīr is from the same root). I do not repeat it here.
What belongs here is where the verse stands. The sentence comes in the middle of a verse that speaks of ingratitude: wa in tashkurū yarḍahu lakum — "if you are thankful, He is pleased with it for you."
And what comes before the sentence is worth recording: wa lā yarḍā li-ʿibādihi'l-kufr — "He is not pleased with ingratitude for His servants."
I record this as an observation: immediately before saying that responsibility is personal, the verse says what it is that is not wanted. Together the two sentences set up a balance: the burden is not shared, but neither is it wanted that anyone should come under it.