Possibly. One of the risks associated with this kind of surgery is damage to the recurrent laryngeal nerve, or RLN.

The RLN runs to the voice box (larynx) from both sides of the neck. RLN injuries can result in hoarseness or even loss of voice from vocal cord paralysis. 

A recent study showed that RLN injuries only happened to 2.7 percent of patients having this kind of surgery for the first time. Most of the injuries cleared up within a few months. In the case of reoperation, though, the risk went up to almost 10 percent.

With this in mind, your surgeon will want to do a very thorough evaluation before deciding how to proceed. Your doctor will use the information to determine which side of the neck will be safest to avoid injuring the nerve.