Do thyroid nodules shrink on their own?

Very unlikely. Thyroid nodules either stay the same or grow. I have never seen one shrink on its own.
They can. Nodules caused by inflammation (thyroiditis) can shrink, or even disappear entirely, when the inflammation resolves. Cystic nodules (fluid filled areas) can shrink if the fluid is reabsorbed by the surrounding tissue. Most nodules, however, either remain the same size they were when discovered, or continue to enlarge.