How do I know if I need to pay a tax on a "gift"?

The company is going to send you a 1099C - cancellation of debt.

You will add this amount to your income and pay tax on it. For example: if you are in the 15% tax bracket, you will owe $503 taxes on this amount ($3353 x .15%)

- this is a very high level analysis. The specific amount of taxes you owe will be dependent on your specific tax situation.

Maybe.

It is not a gift because you received service and the hospital chose not to collect on the bill for the service. If the hospital has billed you and then cancelled the bill, the hospital may send you a 1099C Cancellation of Debt. If this is the case, the amount of cancelled debt is added to your income, and income tax will be owed. If the hospital just never bills you, then there is no taxable event.

that was not a 'gift'

the hospital may issue you a 1099 C on the money they wrote off, and this will be income to you

That's not a gift. It's cancellation of debt, and is normally taxable although there are some exceptions.

No.

If you receive a gift from someone for whom you don't work, then you do not pay tax on it.

You only pay tax on a gift that you get from someone from whom you work (like a Christmas bonus from your employer or the tips that a waiter receives) or a very, very large gift (over $13,000) that you give to someone.

That is not a gift. They forgave the bill. If they send you a 1099misc then you pay taxes on it unless you are insolvent.

There is not tax due from you on this transaction.