Are the side effects of chemo and radiation permanent, or will they eventually go away?

Mostly go away. Side effects will usually improve and go away. However minor liveable changes may be permanent. Often patients may say they are not 100% back to normal but close. It depends on many factors and what has been exposed to how much radiation.
No. The side effects are divided into two parts generally for both radiation and chemotherapy. The short lived side effects: one will experience them during and shortly there after treatment. These side effects go away. The long term side effects: these are delayed and may stay for a long time or permanent.