What effects can losing a kidney have on the body-can it cause fatigue, or affect the immune system?

Single kidney. Loss of one kidney should have no effect on bodily functions. You can live a normal life with one kidney. The one exception would be of you had faling kidneys. In that situation every bit of healthy kidney tissue is needed to maintain kidney functions, so loss of an entire kidney could have a significant consequence. Fatigue and even immune dysfunction could occur with more severe kidney failure.
Nephrectomy. You can have increased immune susceptibility but the body usually compensates over the long term to make up for this loss. And often times it could be the immunosuppressant medications that make you more susceptible to infection.