Here's a 2000 study someone posted on twitter about "Injuries Associated With Bicycle-Towed Child Trailers and Bicycle-Mounted Child Seats" which is the first I've ever seen on the subject.
When compared with bicycle-mounted child seats, there were fewer reported injuries to children associated with the use of bicycle-towed trailers. Motor vehicle involvement and need for hospital admission were similar among injured children in both groups, and the head or face was the most common site of injury. These data imply that ongoing surveillance efforts to identify injuries associated with use of bicycle-towed child trailers are warranted and that bicycle helmets should be worn by children riding in bicycle-towed child trailers and in bicycle-mounted child seats.
the annual injury rate [ related to bicycle-towed child trailers] can be estimated as 0.2 per 100,000 children younger than 5 years. The annual injury rate for bicycle-mounted child seats is estimated at 1.3 per 100,000 children younger than 5 years.
The reasons for more bicycle-mounted injuries are feet in the wheels, and falls from the bike. I feel like with our bike-mounted seat the kids can't get their feet in the spokes if they tried to.
I'd been told that inside the roll cage of a trailer, kids don't really need a helmet. And when I look at the types of injuries kids sustained I can see why I was told that.
Contusions or abrasions were the most frequent injury: this was the diagnosis in 5 (83%) of the injuries related to bicycle-towed child trailers and for 20 (47%) of the injuries associated with the use of bicycle-mounted child seats. Lacerations accounted for 1 (17%) of the injuries among those in bicycle trailers and for 2 (5%) of those in bicycle-mounted child seats. Nine children (21%) with injuries related to bicycle-mounted child seat use had fractures.
As far as I know, helmets aren't designed to prevent contusions, abrasions or lacerations.