I crossed the bridge a few times a couple years ago and never got anything. Personal vehicle not registered in CA.
Rentals are a pain because they will charge a "convenience charge" of approximately $15/day for every day you used the car if you go through one electronic toll booth. Charge will show up about a week after you turn in rental. Electronic tolls are easy for locals but I try to avoid them in rentals.
you need to ask the policy in advance. Avis is who bit me which is frustrating because I like their preferred service otherwise.