Can a goiter grow on only one side of the neck?

Yes. Goiter is simply the term for an enlarged thyroid. Usually it is bilateral (both sides) but you can have marked difference from one side or the other. One sided masses of the thyroid are more concerning for a thyroid cancer, and so should be evaluated at least with an ultrasound, and perhaps with a needle biopsy.
Yes. Can be in either or both lobes of the thyroid gland. A thyroid ultrasound should be able to distinguish and determine if there is any underlying malignancy or cystic disease.