Is this safe? What are the risks and/or side effects?
I don't think you have a varicose vein on your face...? That would be a very large, puffy, discolored vein. There are often small reticular veins on the nose and under the eyes. Depending on skin color these veins can show more. They can absolutely be effectively treated with sclerotherapy and it is very safe, assuming you are seeing a good physician injector. There are some people who caution against sclerotherapy for the face but when the injecting solution is diluted more, there is nothing better. I have done it for nearly 30 years. "This answer has been solicited without seeing this patient and cannot be held as true medical advice, but only opinion. Seek in-person treatment with a trained medical professional for appropriate care."