I have a car seat that was loaned to a family member and returned with the internal styrofoam structure (under the fabric) broken on one side (see picture)
Is the seat safe to use or do I need to get a new one?
No. I would not risk it. Usually the manual will say not to use the unit if it's damaged, or even to use it if it's older than a few years. Parts may look just little bit damaged, but that little bit can cause a failure during an accident that will leave your little one hurt. Of course, like everything else, there's a tradeoff between cost (in this case free) and safety, but I personally would never use a damaged safety seat. Save money elsewhere, like by buying less plastic toys they get tired of after a day. :-)