"Saute" is simply a fancy word which means to fry lightly in fat in a shallow open pan.
After the chicken is sauted, use a slotted spoon or tongs to remove the chicken from the pan (i.e. you only want to remove the meat and none of the juices that are left in the pan). Place the cooked chicken onto the paper towels so that the excess fat is drained off of the cooked meat.
The reason they have you put the vegetables into the same pan with the oil previously used to cook the chicken is because the oils and chicken flavors left there will nicely flavor the next ingrediants.
You don't have to worry about transferring any harmful bacteria to the vegetables because cooking the chicken thoroughly removes those bacteria! The only transferring going on will be flavors. =) So as long as your meat is cooked to the proper doneness you don't have anything to worry about.
Make sure you wash your cutting board and knives thoroughly after cutting meat ~ that's the most likely place where any possible contamination would take place.