Private student loans don't go away unless you pay them off, but in most cases, they'll fall off your credit report after seven years. But keep in mind that lenders can still contact you to collect an old debt, even if it's decades old and they can no longer take you to court over it.
While there still are some questions, there's one thing we know for sure: The Biden administration is not canceling private student loans. The student loan forgiveness announced by the Biden administration applies only to those with federal student loans.