Your child wanted to buy a car and couldn't get approved alone. You decided to co-sign, saying you'd share the car. The total payment was $500 per month, and you both paid $250 of that. If your child then goes bankrupt, do you still have to pay for the debt, or is it all cleared since it was the same auto loan?
Every case is different, but you generally have to pay. You still said that you'd pay off that debt, and, even though your child can't do it, the lender doesn't care. One of you has to pay, and they can come after you for the full amount.
When you co-sign, it's wise to assume that you're signing on all by yourself. If you can afford the payments alone, co-signing may be a way to lower them. But you should always have the idea in the back of your mind that you could be responsible for the entire debt.
That being said, some lenders understand the situation and won't demand that you pay it all instantly. Maybe you could only afford $250 per month and there's no way you can pay $500. They may just let you keep paying $250. Though it will take longer, you'll eventually pay off everything. Lenders aren't guaranteed to do this, but they will in some cases. After all, they'd rather get the payments over a longer period of time than not get them at all.
As you can see, you must understand the implications of any financial contract. Be sure you know how bankruptcy works, what it could mean for that contract, and what rights and obligations you have.
Source: eCheck, "How to Declare Bankruptcy In The USA," accessed March 31, 2017