More info needed. You need more information. Are the children ill? Is the babysitter ill? What is causing the enlarged lymph nodes?
Not usually. It is unlikely that there is a transmittable cause for this, but without knowing why the other child's lymph nodes are large it is difficult to answer your question. The babysitter may get infectious mononucleosis (eb virus) from the other child and then transmit it to your child, but this is unlikely and without knowing what the other child had, difficult to speculate on.