I worked as a ped med asst for my kids' pediatric office before becoming an RN. At that time, there were only 2 RNs and the rest of us were med assts. The RNs worked more as supervisors and had less patient contact than we did. Eventually, they were let go. In my state (AZ), most private practices are not hiring RNs because they do not want to pay their higher salaries.
What I did (all under the direction of a doctor):
well visits - heights/weights for older kiddos, length/head circumference/weight on infants, vitals, review vaccine histories & admin vaccines, vision and hearing tests, toe pokes to check iron levels on infants, heel sticks for PKU or bilirubin levels on newborns, set up sterile fields and assist on circumcisions.
sick visits - take histories, vitals, strep tests, urinalysis, pulse ox & admin breathing treatments, ear & eye lavages, irrigate/cleanse wounds, put splints on fingers, admin IM abx.
You also assist the Dr anytime they need help calming/holding a scared kiddo during suture removal or even just looking in the kiddo's ear! There are also clerical duties such as faxing lab or xray slips, calling parents with results, calling in RXs to pharms.
Overall, working in a private ped practice is great - lots of fun! However, as I stated earlier and at least in my state private practice is phasing out RNs. Best of luck to you if that is the direction you are headed!