Your hesitancy is understandable, and it's crucial to have clear facts. The chickenpox vaccine does contain a live virus, but it is a significantly weakened (attenuated) form of the wild varicella-zoster virus. This is a key distinction. Because it is weakened, it stimulates the immune system to produce protective antibodies without causing the full-blown illness in a healthy individual.
In very rare cases (about 1-5% of recipients), a person may develop a very mild rash with a few bumps days or weeks after the shot. This is not a true case of chickenpox but a side effect of the vaccine working to build immunity. It is also minimally contagious from this rash.
Regarding side effects:
Most Common: These are typical of many vaccines and include soreness, redness, or swelling at the injection site, a mild fever, and temporary fatigue.
Serious Side Effects: These are exceedingly rare. The risk of a severe allergic reaction (anaphylaxis) is less than 1 in a million doses. There is a theoretical risk of the virus spreading in the body, but this has only been reported in a handful of cases, almost exclusively in children with previously unknown, severe weaknesses in their immune systems.
For the vast majority of people, the known risks of the actual chickenpox disease—which include hospitalization from bacterial skin infections, pneumonia, and encephalitis—are far greater than the risks associated with the vaccine.