What are common Fraxel side effects?

To be precise, side effects are expected reactions from a treatment that are not complications. You ask about Fraxel. Side effects for Fraxel RE:store are different than Fraxel Re:pair. Reactions to Fraxel restore include swelling, redness and gentle skin flaking for three to five days. Sometimes the skin flakes slightly tan or brown. There is stinging during the treatment and for several minutes to even an hour or more after the procedure rarely. There can be a reactivation of one's susceptibility to having cold sores, and there can be pigmentation development that usually is temporary.
Fraxel Repair is more aggressive as a treatment and raw skin occurs as a result so that side effects are more numerable including oozing, crusting, brown flaking, long term redness, sensitivity and a very small risk of scarring and pigmentation changes.

It depends on how aggressive your treatment is. For lighter treatments, you might experience some mild swelling, redness, and tenderness for a few days. More extensive treatments might be accompanied by redness, swelling, and flaking, which could require a few days of social downtime before you feel ready to appear in public. These side effects are expected after the treatment and will resolve without intervention in nearly all cases. Make sure you work with an experienced, board-certified dermatologist with expertise in the Fraxel devices to ensure the best results with minimal risks.

Fraxel laser treatments are very safe.  I don't know which treatment you  are considering.  Fraxel is a brand name of fractionated laser treatments.  There are both ablative and non-ablative treatments meaning that some ablate (burn off) the outer layer of skin and some don't.  Restore is Fraxel's non-ablative treatment and Repair is their ablative laser. 

So, what can go wrong?  Burns are extremely rare in experienced hands.  Special care must be taken in dark skinned individuals to reduce the risk of hyperpigmentation.  Individuals who get cold sores should take preventative medicine.  The doctor can prescibe this.