Do you get taxed on money that's been left to you come tax time

Not unless the government actually owes you any... from tax overpayments.

They don't run Trust funds.

Taxes on inheritance are generally paid by the estate. There is also a federal exclusion of 2 million, that most states adopt.

Ask the trustee. Someone pays taxes on the income earned on the trust funds. You will get a K1 if you pay it. Otherwise the trust pays it.